CINXE.COM

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <title>On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion | Advances in Continuous and Discrete Models | Full Text</title> <meta name="citation_abstract" content="In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study."/> <meta name="journal_id" content="13662"/> <meta name="dc.title" content="On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion"/> <meta name="dc.source" content="Advances in Difference Equations 2016 2016:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="SpringerOpen"/> <meta name="dc.date" content="2016-07-23"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2016 Pei and Xu"/> <meta name="dc.rights" content="2016 Pei and Xu"/> <meta name="dc.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="dc.description" content="In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study."/> <meta name="prism.issn" content="1687-1847"/> <meta name="prism.publicationName" content="Advances in Difference Equations"/> <meta name="prism.publicationDate" content="2016-07-23"/> <meta name="prism.volume" content="2016"/> <meta name="prism.number" content="1"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="15"/> <meta name="prism.copyright" content="2016 Pei and Xu"/> <meta name="prism.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="prism.url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-016-0916-1"/> <meta name="prism.doi" content="doi:10.1186/s13662-016-0916-1"/> <meta name="citation_pdf_url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-016-0916-1"/> <meta name="citation_fulltext_html_url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-016-0916-1"/> <meta name="citation_journal_title" content="Advances in Difference Equations"/> <meta name="citation_journal_abbrev" content="Adv Differ Equ"/> <meta name="citation_publisher" content="SpringerOpen"/> <meta name="citation_issn" content="1687-1847"/> <meta name="citation_title" content="On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion"/> <meta name="citation_volume" content="2016"/> <meta name="citation_issue" content="1"/> <meta name="citation_publication_date" content="2016/12"/> <meta name="citation_online_date" content="2016/07/23"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="15"/> <meta name="citation_article_type" content="Research"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1186/s13662-016-0916-1"/> <meta name="DOI" content="10.1186/s13662-016-0916-1"/> <meta name="size" content="705067"/> <meta name="citation_doi" content="10.1186/s13662-016-0916-1"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1186/s13662-016-0916-1&api_key="/> <meta name="description" content="In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study."/> <meta name="dc.creator" content="Pei, Bin"/> <meta name="dc.creator" content="Xu, Yong"/> <meta name="dc.subject" content="Difference and Functional Equations"/> <meta name="dc.subject" content="Mathematics, general"/> <meta name="dc.subject" content="Analysis"/> <meta name="dc.subject" content="Functional Analysis"/> <meta name="dc.subject" content="Ordinary Differential Equations"/> <meta name="dc.subject" content="Partial Differential Equations"/> <meta name="citation_reference" content="citation_title=Stochastic Differential Equations; citation_publication_date=2005; citation_id=CR1; citation_author=B Øksendal; citation_publisher=Springer"/> <meta name="citation_reference" content="citation_title=Stochastic Differential Equations, Theory and Applications; citation_publication_date=1974; citation_id=CR2; citation_author=L Arnold; citation_publisher=John Wiley and Sons"/> <meta name="citation_reference" content="citation_title=Stochastic Differential Equations and Applications; citation_publication_date=2006; citation_id=CR3; citation_author=A Friedman; citation_publisher=Dover Publications"/> <meta name="citation_reference" content="citation_title=Introduction to Stochastic Differential Equations; citation_publication_date=1988; citation_id=CR4; citation_author=T Gard; citation_publisher=Marcel Dekker"/> <meta name="citation_reference" content="citation_journal_title=Trans. Am. Soc. Civ. Eng.; citation_title=Long-term storage capacity in reservoirs; citation_author=H Hurst; citation_volume=116; citation_publication_date=1951; citation_pages=400-410; citation_id=CR5"/> <meta name="citation_reference" content="citation_journal_title=C. R. (Dokl.) Acad. Sci. URSS; citation_title=Wienersche spiralen und einige andere interessante kurven im Hilbertschen raum; citation_author=A Kolmogorov; citation_volume=26; citation_publication_date=1940; citation_pages=115-118; citation_id=CR6"/> <meta name="citation_reference" content="citation_journal_title=SIAM Rev.; citation_title=Fractional Brownian motions, fractional noises and applications; citation_author=B Mandelbrot, J Ness; citation_volume=10; citation_issue=4; citation_publication_date=1968; citation_pages=422-427; citation_doi=10.1137/1010093; citation_id=CR7"/> <meta name="citation_reference" content="citation_journal_title=Chem. Phys. Lett.; citation_title=Fractional Brownian motion models for polymers; citation_author=N Chakravarti, K Sebastian; citation_volume=267; citation_publication_date=1997; citation_pages=9-13; citation_doi=10.1016/S0009-2614(97)00075-4; citation_id=CR8"/> <meta name="citation_reference" content="citation_journal_title=Infin. Dimens. Anal. Quantum Probab. Relat. Top.; citation_title=Fractional white noise calculus and application to finance; citation_author=Y Hu, B Øksendal; citation_volume=6; citation_publication_date=2003; citation_pages=1-32; citation_doi=10.1142/S0219025703001110; citation_id=CR9"/> <meta name="citation_reference" content="citation_journal_title=Chem. Eng. Sci.; citation_title=The fractional Brownian motion as a model for an industrial airlift reactor; citation_author=R Scheffer, F Maciel; citation_volume=56; citation_publication_date=2001; citation_pages=707-711; citation_doi=10.1016/S0009-2509(00)00279-7; citation_id=CR10"/> <meta name="citation_reference" content="citation_journal_title=Rev. Mat. Iberoam.; citation_title=Differential equations driven by rough signals; citation_author=T Lyons; citation_volume=14; citation_publication_date=1998; citation_pages=215-310; citation_doi=10.4171/RMI/240; citation_id=CR11"/> <meta name="citation_reference" content="citation_journal_title=Collect. Math.; citation_title=Differential equations driven by fractional Brownian motion; citation_author=D Nualart, A Rascanu; citation_volume=53; citation_publication_date=2002; citation_pages=55-81; citation_id=CR12"/> <meta name="citation_reference" content="citation_title=Stochastic Calculus for Fractional Brownian Motion and Related Processes; citation_publication_date=2008; citation_id=CR13; citation_author=Y Mishura; citation_publisher=Springer"/> <meta name="citation_reference" content="citation_title=Stochastic Calculus for Fractional Brownian Motion and Applications; citation_publication_date=2008; citation_id=CR14; citation_author=F Biagini; citation_author=Y Hu; citation_author=B Oksendal; citation_author=T Zhang; citation_publisher=Springer"/> <meta name="citation_reference" content="citation_journal_title=Econometrica; citation_title=A theory of the term structure of interest rate; citation_author=J Cox, J Ingersoll, S Ross; citation_volume=53; citation_publication_date=1985; citation_pages=385-407; citation_doi=10.2307/1911242; citation_id=CR15"/> <meta name="citation_reference" content="citation_journal_title=J. Differ. Equ.; citation_title=Successive approximations to solutions of stochastic differential equations; citation_author=T Taniguchi; citation_volume=96; citation_publication_date=1992; citation_pages=152-169; citation_doi=10.1016/0022-0396(92)90148-G; citation_id=CR16"/> <meta name="citation_reference" content="citation_journal_title=Int. J. Theor. Appl. Finance; citation_title=Pricing multi-asset options with an external barrier; citation_author=Y Kwok; citation_volume=1; citation_publication_date=1998; citation_pages=523-541; citation_doi=10.1142/S021902499800028X; citation_id=CR17"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Kyoto Univ.; citation_title=On the uniqueness of solution of stochastic differential equations II; citation_author=S Watanabe, T Yamada; citation_volume=11; citation_issue=3; citation_publication_date=1971; citation_pages=553-563; citation_id=CR18"/> <meta name="citation_reference" content="citation_journal_title=J. Lond. Math. Soc.; citation_title=One dimensional stochastic differential equations with no strong solution; citation_author=M Barlow; citation_volume=26; citation_publication_date=1982; citation_pages=335-347; citation_doi=10.1112/jlms/s2-26.2.335; citation_id=CR19"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Kyoto Univ.; citation_title=On a comparison theorem for solutions of stochastic differential equations and its applications; citation_author=T Yamada; citation_volume=13; citation_issue=3; citation_publication_date=1973; citation_pages=497-512; citation_id=CR20"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Kyoto Univ.; citation_title=On the successive approximation of solutions of stochastic differential equations; citation_author=T Yamada; citation_volume=21; citation_issue=3; citation_publication_date=1981; citation_pages=501-515; citation_id=CR21"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Kyoto Univ.; citation_title=On the uniqueness of solutions of stochastic differential equations; citation_author=T Yamada, S Watanabe; citation_volume=11; citation_publication_date=1971; citation_pages=155-167; citation_id=CR22"/> <meta name="citation_reference" content="citation_journal_title=Ann. Inst. Henri Poincaré Probab. Stat.; citation_title=Stochastic integration with respect to fractional Brownian motion; citation_author=P Carmona, L Coutin, G Montseny; citation_volume=39; citation_publication_date=2003; citation_pages=27-68; citation_doi=10.1016/S0246-0203(02)01111-1; citation_id=CR23"/> <meta name="citation_reference" content="citation_journal_title=Ann. Probab.; citation_title=Stochastic calculus with respect to Gaussian process; citation_author=E Alòs, O Mazet, D Nualart; citation_volume=29; citation_publication_date=2001; citation_pages=766-801; citation_doi=10.1214/aop/1008956692; citation_id=CR24"/> <meta name="citation_reference" content="citation_journal_title=SIAM J. Control Optim.; citation_title=Stochastic calculus for fractional Brownian motion I: theory; citation_author=T Duncan, Y Hu, B Pasik-Duncan; citation_volume=38; citation_publication_date=2000; citation_pages=582-612; citation_doi=10.1137/S036301299834171X; citation_id=CR25"/> <meta name="citation_reference" content="citation_journal_title=Stoch. Stoch. Rep.; citation_title=Stochastic integration with respect to the fractional Brownian motion; citation_author=E Alòs, D Nualart; citation_volume=75; citation_issue=3; citation_publication_date=2003; citation_pages=129-152; citation_doi=10.1080/1045112031000078917; citation_id=CR26"/> <meta name="citation_reference" content="citation_journal_title=Probab. Theory Relat. Fields; citation_title=Forward, backward and symmetric stochastic integration; citation_author=F Russo, P Vallois; citation_volume=97; citation_publication_date=1993; citation_pages=403-421; citation_doi=10.1007/BF01195073; citation_id=CR27"/> <meta name="citation_reference" content="citation_journal_title=Discrete Contin. Dyn. Syst., Ser. B; citation_title=Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion; citation_author=Y Xu, B Pei, R Guo; citation_volume=20; citation_publication_date=2015; citation_pages=2257-2267; citation_doi=10.3934/dcdsb.2015.20.2257; citation_id=CR28"/> <meta name="citation_reference" content="citation_journal_title=Discrete Contin. Dyn. Syst., Ser. B; citation_title=Stochastic averaging principle for dynamical systems with fractional Brownian motion; citation_author=Y Xu, R Guo; citation_volume=19; citation_issue=4; citation_publication_date=2014; citation_pages=1197-1212; citation_doi=10.3934/dcdsb.2014.19.1197; citation_id=CR29"/> <meta name="citation_reference" content="citation_journal_title=Abstr. Appl. Anal.; citation_title=An averaging principle for stochastic differential delay equations with fractional Brownian motion; citation_author=Y Xu, B Pei, Y Li; citation_volume=2014; citation_publication_date=2014; citation_id=CR30"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Anal. Appl.; citation_title=Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients; citation_author=S Albeverio, Z Brzézniak, J Wu; citation_volume=371; citation_publication_date=2010; citation_pages=309-322; citation_doi=10.1016/j.jmaa.2010.05.039; citation_id=CR31"/> <meta name="citation_reference" content="citation_journal_title=J. Math. Anal. Appl.; citation_title=The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations; citation_author=T Taniguchi; citation_volume=360; citation_publication_date=2009; citation_pages=245-253; citation_doi=10.1016/j.jmaa.2009.06.007; citation_id=CR32"/> <meta name="citation_reference" content="citation_journal_title=Czechoslov. Math. J.; citation_title=Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients; citation_author=D Barbu, G Bocsan; citation_volume=52; citation_publication_date=2002; citation_pages=87-95; citation_doi=10.1023/A:1021723421437; citation_id=CR33"/> <meta name="citation_reference" content="citation_journal_title=Stoch. Dyn.; citation_title=Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion; citation_author=Y Xu, B Pei, J Wu; citation_publication_date=2016; citation_id=CR34"/> <meta name="citation_reference" content="citation_journal_title=Appl. Math. Comput.; citation_title=Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise; citation_author=Y Xu, B Pei, G Guo; citation_volume=263; citation_publication_date=2015; citation_pages=398-409; citation_id=CR35"/> <meta name="citation_author" content="Pei, Bin"/> <meta name="citation_author_institution" content="Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China"/> <meta name="citation_author" content="Xu, Yong"/> <meta name="citation_author_institution" content="Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/static/img/favicons/darwin/apple-touch-icon.png> <link rel="icon" type="image/png" sizes="192x192" href=/static/img/favicons/darwin/android-chrome-192x192.png> <link rel="icon" type="image/png" sizes="32x32" href=/static/img/favicons/darwin/favicon-32x32.png> <link rel="icon" type="image/png" sizes="16x16" href=/static/img/favicons/darwin/favicon-16x16.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/static/img/favicons/darwin/favicon.ico> <meta name="theme-color" content="#e6e6e6"> <script>(function(H){H.className=H.className.replace(/\bno-js\b/,'js')})(document.documentElement)</script> <link rel="stylesheet" media="screen" href=/static/app-springeropen/css/core-article-f3872e738d.css> <link rel="stylesheet" media="screen" href=/static/app-springeropen/css/core-b516af10bc.css> <link rel="stylesheet" media="print" href=/static/app-springeropen/css/print-b8af42253b.css>  <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { button{line-height:inherit}html,label{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif}html{-webkit-font-smoothing:subpixel-antialiased;box-sizing:border-box;color:#333;font-size:100%;height:100%;line-height:1.61803;overflow-y:scroll}*{box-sizing:inherit}body{background:#fcfcfc;margin:0;max-width:100%;min-height:100%}button,div,form,input,p{margin:0;padding:0}body{padding:0}a{color:#004b83;text-decoration:underline;text-decoration-skip-ink:auto}a>img{vertical-align:middle}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h3{font-family:Georgia,Palatino,serif;font-style:normal;font-weight:400;line-height:1.4}h3{font-size:1.5rem}h1,h2,h3{margin:0}h2+*{margin-block-start:1rem}h1+*{margin-block-start:3rem}[style*="display: none"]:first-child+*{margin-block-start:0}.c-navbar{background:#e6e6e6;border-bottom:1px solid #d9d9d9;border-top:1px solid #d9d9d9;font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;line-height:1.61803;padding:16px 0}.c-navbar--with-submit-button{padding-bottom:24px}@media only screen and (min-width:540px){.c-navbar--with-submit-button{padding-bottom:16px}}.c-navbar__container{display:flex;flex-wrap:wrap;justify-content:space-between;margin:0 auto;max-width:1280px;padding:0 16px}.c-navbar__content{display:flex;flex:0 1 auto}.c-navbar__nav{align-items:center;display:flex;flex-wrap:wrap;gap:16px 16px;list-style:none;margin:0;padding:0}.c-navbar__item{flex:0 0 auto}.c-navbar__link{background:0 0;border:0;color:currentcolor;display:block;text-decoration:none;text-transform:capitalize}.c-navbar__link--is-shown{text-decoration:underline}.c-ad{text-align:center}@media only screen and (min-width:320px){.c-ad{padding:8px}}.c-ad--728x90{background-color:#ccc;display:none}.c-ad--728x90 .c-ad__inner{min-height:calc(1.5em + 94px)}.c-ad--728x90 iframe{height:90px;max-width:970px}@media only screen and (min-width:768px){.js .c-ad--728x90{display:none}.js .u-show-following-ad+.c-ad--728x90{display:block}}.c-ad iframe{border:0;overflow:auto;vertical-align:top}.c-ad__label{color:#333;font-weight:400;line-height:1.5;margin-bottom:4px}.c-ad__label,.c-skip-link{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:.875rem}.c-skip-link{background:#f7fbfe;bottom:auto;color:#004b83;padding:8px;position:absolute;text-align:center;transform:translateY(-100%);z-index:9999}@media (prefers-reduced-motion:reduce){.c-skip-link{transition:top .3s ease-in-out 0s}}@media print{.c-skip-link{display:none}}.c-skip-link:link{color:#004b83}.c-dropdown__button:after{border-color:transparent transparent transparent #fff;border-style:solid;border-width:4px 0 4px 14px;content:"";display:block;height:0;margin-left:3px;width:0}.c-dropdown{display:inline-block;position:relative}.c-dropdown__button{background-color:transparent;border:0;display:inline-block;padding:0;white-space:nowrap}.c-dropdown__button:after{border-color:currentcolor transparent transparent;border-width:5px 4px 0 5px;display:inline-block;margin-left:8px;vertical-align:middle}.c-dropdown__menu{background-color:#fff;border:1px solid #d9d9d9;border-radius:3px;box-shadow:0 2px 6px rgba(0,0,0,.1);font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:1.125rem;line-height:1.4;list-style:none;margin:0;padding:8px 0;position:absolute;top:100%;transform:translateY(8px);width:180px;z-index:100}.c-dropdown__menu:after,.c-dropdown__menu:before{border-style:solid;bottom:100%;content:"";display:block;height:0;left:16px;position:absolute;width:0}.c-dropdown__menu:before{border-color:transparent transparent #d9d9d9;border-width:0 9px 9px;transform:translateX(-1px)}.c-dropdown__menu:after{border-color:transparent transparent #fff;border-width:0 8px 8px}.c-dropdown__menu--right{left:auto;right:0}.c-dropdown__menu--right:after,.c-dropdown__menu--right:before{left:auto;right:16px}.c-dropdown__menu--right:before{transform:translateX(1px)}.c-dropdown__link{background-color:transparent;color:#004b83;display:block;padding:4px 16px}.c-header{background-color:#fff;border-bottom:4px solid #00285a;font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:1.125rem;padding:16px 0}.c-header__container,.c-header__menu{align-items:center;display:flex;flex-wrap:wrap}@supports (gap:2em){.c-header__container,.c-header__menu{gap:2em 2em}}.c-header__menu{list-style:none;margin:0;padding:0}.c-header__item{color:inherit}@supports not (gap:2em){.c-header__item{margin-left:24px}}.c-header__container{justify-content:space-between;margin:0 auto;max-width:1280px;padding:0 16px}@supports not (gap:2em){.c-header__brand{margin-right:32px}}.c-header__brand a{display:block;text-decoration:none}.c-header__link{color:inherit}.c-form-field{margin-bottom:1em}.c-form-field__label{color:#666;display:block;font-size:.875rem;margin-bottom:.4em}.c-form-field__input{border:1px solid #b3b3b3;border-radius:3px;box-shadow:inset 0 1px 3px 0 rgba(0,0,0,.21);font-size:.875rem;line-height:1.28571;padding:.75em 1em;vertical-align:middle;width:100%}.c-journal-header__title>a{color:inherit}.c-popup-search{background-color:#f2f2f2;box-shadow:0 3px 3px -3px rgba(0,0,0,.21);padding:16px 0;position:relative;z-index:10}@media only screen and (min-width:1024px){.js .c-popup-search{position:absolute;top:100%;width:100%}.c-popup-search__container{margin:auto;max-width:70%}}.ctx-search .c-form-field{margin-bottom:0}.ctx-search .c-form-field__input{border-bottom-right-radius:0;border-top-right-radius:0;margin-right:0}.c-journal-header{background-color:#f2f2f2;padding-top:16px}.c-journal-header__title{font-size:1.3125rem;margin:0 0 16px}.c-journal-header__grid{column-gap:1.25rem;display:grid;grid-template-areas:"main" "side";grid-template-columns:1fr;width:100%}@media only screen and (min-width:768px){.c-journal-header__grid{column-gap:1.25rem;grid-template-areas:"main side";grid-template-columns:1fr 160px}}@media only screen and (min-width:1024px){.c-journal-header__grid{column-gap:3.125rem;grid-template-areas:"main side";grid-template-columns:1fr 190px}}@media only screen and (min-width:768px){.c-journal-header__grid-main{margin:0!important;width:auto!important}}.c-journal-header__grid-main{grid-area:main/main/main/main}.c-navbar{font-size:.875rem}.u-button{align-items:center;background-color:#f2f2f2;background-image:linear-gradient(#fff,#f2f2f2);border:1px solid #ccc;border-radius:2px;cursor:pointer;display:inline-flex;font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:1rem;justify-content:center;line-height:1.3;margin:0;padding:8px;position:relative;text-decoration:none;transition:all .25s ease 0s,color .25s ease 0s,border-color .25s ease 0s;width:auto}.u-button svg,.u-button--primary svg,.u-button--tertiary svg{fill:currentcolor}.u-button{color:#004b83}.u-button--primary,.u-button--tertiary{background-color:#33629d;background-image:linear-gradient(#4d76a9,#33629d);border:1px solid rgba(0,59,132,.5);color:#fff}.u-button--tertiary{font-weight:400}.u-button--full-width{display:flex;width:100%}.u-clearfix:after,.u-clearfix:before{content:"";display:table}.u-clearfix:after{clear:both}.u-color-open-access{color:#b74616}.u-container{margin:0 auto;max-width:1280px;padding:0 16px}.u-display-flex{display:flex;width:100%}.u-align-items-center{align-items:center}.u-justify-content-space-between{justify-content:space-between}.u-flex-static{flex:0 0 auto}.u-display-none{display:none}.js .u-js-hide{display:none;visibility:hidden}@media print{.u-hide-print{display:none}}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.u-position-relative{position:relative}.u-mt-32{margin-top:32px}.u-mr-24{margin-right:24px}.u-mr-48{margin-right:48px}.u-mb-32{margin-bottom:32px}.u-ml-8{margin-left:8px}.u-button-reset{background-color:transparent;border:0;padding:0}.u-text-sm{font-size:1rem}.u-h3,.u-h4{font-style:normal;line-height:1.4}.u-h3{font-family:Georgia,Palatino,serif;font-size:1.5rem;font-weight:400}.u-h4{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:1.25rem;font-weight:700}.u-vh-full{min-height:100vh}.u-hide{display:none;visibility:hidden}.u-hide:first-child+*{margin-block-start:0}@media only screen and (min-width:1024px){.u-hide-at-lg{display:none;visibility:hidden}}@media only screen and (max-width:1023px){.u-hide-at-lt-lg{display:none;visibility:hidden}.u-hide-at-lt-lg:first-child+*{margin-block-start:0}}.u-visually-hidden{clip:rect(0,0,0,0);border:0;height:1px;margin:-100%;overflow:hidden;padding:0;position:absolute!important;width:1px}.u-button--tertiary{font-size:.875rem;padding:8px 16px}@media only screen and (max-width:539px){.u-button--alt-colour-on-mobile{background-color:#f2f2f2;background-image:linear-gradient(#fff,#f2f2f2);border:1px solid #ccc;color:#004b83}}body{font-size:1.125rem}.c-header__navigation{display:flex;gap:.5rem .5rem} }</style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { button{line-height:inherit}html,label{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif}html{-webkit-font-smoothing:subpixel-antialiased;box-sizing:border-box;color:#333;font-size:100%;height:100%;line-height:1.61803;overflow-y:scroll}*{box-sizing:inherit}body{background:#fcfcfc;margin:0;max-width:100%;min-height:100%}button,div,form,input,p{margin:0;padding:0}body{padding:0}a{color:#004b83;overflow-wrap:break-word;text-decoration:underline;text-decoration-skip-ink:auto;word-break:break-word}a>img{vertical-align:middle}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h3{font-family:Georgia,Palatino,serif;font-style:normal;font-weight:400;line-height:1.4}h3{font-size:1.5rem}h1,h2,h3{margin:0}h2+*{margin-block-start:1rem}h1+*{margin-block-start:3rem}[style*="display: none"]:first-child+*{margin-block-start:0}p{overflow-wrap:break-word;word-break:break-word}.c-article-associated-content__container .c-article-associated-content__collection-label,.u-h3{font-weight:700}.u-h3{font-size:1.5rem}.c-reading-companion__figure-title,.u-h4{font-size:1.25rem;font-weight:700}body{font-size:1.125rem}.c-article-header{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;margin-bottom:40px}.c-article-identifiers{color:#6f6f6f;display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3;list-style:none;margin:0 0 8px;padding:0}.c-article-identifiers__item{border-right:1px solid #6f6f6f;list-style:none;margin-right:8px;padding-right:8px}.c-article-identifiers__item:last-child{border-right:0;margin-right:0;padding-right:0}.c-article-title{font-size:1.5rem;line-height:1.25;margin-bottom:16px}@media only screen and (min-width:768px){.c-article-title{font-size:1.875rem;line-height:1.2}}.c-article-author-list{display:inline;font-size:1rem;list-style:none;margin:0 8px 0 0;padding:0;width:100%}.c-article-author-list__item{display:inline;padding-right:0}.c-article-author-list svg{margin-left:4px}.c-article-author-list__show-more{display:none;margin-right:4px}.c-article-author-list__button,.js .c-article-author-list__item--hide,.js .c-article-author-list__show-more{display:none}.js .c-article-author-list--long .c-article-author-list__show-more,.js .c-article-author-list--long+.c-article-author-list__button{display:inline}@media only screen and (max-width:539px){.js .c-article-author-list__item--hide-small-screen{display:none}.js .c-article-author-list--short .c-article-author-list__show-more,.js .c-article-author-list--short+.c-article-author-list__button{display:inline}}#uptodate-client,.js .c-article-author-list--expanded .c-article-author-list__show-more{display:none!important}.js .c-article-author-list--expanded .c-article-author-list__item--hide-small-screen{display:inline!important}.c-article-author-list__button,.c-button-author-list{background:#ebf1f5;border:4px solid #ebf1f5;border-radius:20px;color:#666;font-size:.875rem;line-height:1.4;padding:2px 11px 2px 8px;text-decoration:none}.c-article-author-list__button svg,.c-button-author-list svg{margin:1px 4px 0 0}.c-article-author-list__button:hover,.c-button-author-list:hover{background:#173962;border-color:transparent;color:#fff}.c-article-info-details{font-size:1rem;margin-bottom:8px;margin-top:16px}.c-article-info-details__cite-as{border-left:1px solid #6f6f6f;margin-left:8px;padding-left:8px}.c-article-metrics-bar{display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3}.c-article-metrics-bar__wrapper{margin:0 0 16px}.c-article-metrics-bar__item{align-items:baseline;border-right:1px solid #6f6f6f;margin-right:8px}.c-article-metrics-bar__item:last-child{border-right:0}.c-article-metrics-bar__count{font-weight:700;margin:0}.c-article-metrics-bar__label{color:#626262;font-style:normal;font-weight:400;margin:0 10px 0 5px}.c-article-metrics-bar__details{margin:0}.c-article-main-column{font-family:Georgia,Palatino,serif;margin-right:8.6%;width:60.2%}@media only screen and (max-width:1023px){.c-article-main-column{margin-right:0;width:100%}}.c-article-extras{float:left;font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;width:31.2%}@media only screen and (max-width:1023px){.c-article-extras{display:none}}.c-article-associated-content__container .c-article-associated-content__title,.c-article-section__title{border-bottom:2px solid #d5d5d5;font-size:1.25rem;margin:0;padding-bottom:8px}@media only screen and (min-width:768px){.c-article-associated-content__container .c-article-associated-content__title,.c-article-section__title{font-size:1.5rem;line-height:1.24}}.c-article-associated-content__container .c-article-associated-content__title{margin-bottom:8px}.c-article-section{clear:both}.c-article-section__content{margin-bottom:40px;margin-top:0;padding-top:8px}@media only screen and (max-width:1023px){.c-article-section__content{padding-left:0}}.c-article__sub-heading{color:#222;font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif;font-size:1.125rem;font-style:normal;font-weight:400;line-height:1.3;margin:24px 0 8px}@media only screen and (min-width:768px){.c-article__sub-heading{font-size:1.5rem;line-height:1.24}}.c-article__sub-heading:first-child{margin-top:0}.c-article-authors-search{margin-bottom:24px;margin-top:0}.c-article-authors-search__item,.c-article-authors-search__title{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Oxygen-Sans,Ubuntu,Cantarell,Helvetica Neue,sans-serif}.c-article-authors-search__title{color:#626262;font-size:1.05rem;font-weight:700;margin:0;padding:0}.c-article-authors-search__item{font-size:1rem}.c-article-authors-search__text{margin:0}.c-article-share-box__no-sharelink-info{font-size:.813rem;font-weight:700;margin-bottom:24px;padding-top:4px}.c-article-share-box__only-read-input{border:1px solid #d5d5d5;box-sizing:content-box;display:inline-block;font-size:.875rem;font-weight:700;height:24px;margin-bottom:8px;padding:8px 10px}.c-article-share-box__button--link-like{background-color:transparent;border:0;color:#069;cursor:pointer;font-size:.875rem;margin-bottom:8px;margin-left:10px}.c-article-associated-content__container .c-article-associated-content__collection-label{font-size:.875rem;line-height:1.4}.c-article-associated-content__container .c-article-associated-content__collection-title{line-height:1.3}.c-context-bar{box-shadow:0 0 10px 0 rgba(51,51,51,.2);position:relative;width:100%}.c-context-bar__title{display:none}.c-reading-companion{clear:both;min-height:389px}.c-reading-companion__sticky{max-width:389px}.c-reading-companion__scroll-pane{margin:0;min-height:200px;overflow:hidden auto}.c-reading-companion__tabs{display:flex;flex-flow:row nowrap;font-size:1rem;list-style:none;margin:0 0 8px;padding:0}.c-reading-companion__tabs>li{flex-grow:1}.c-reading-companion__tab{background-color:#eee;border:1px solid #d5d5d5;border-image:initial;border-left-width:0;color:#069;font-size:1rem;padding:8px 8px 8px 15px;text-align:left;width:100%}.c-reading-companion__tabs li:first-child .c-reading-companion__tab{border-left-width:1px}.c-reading-companion__tab--active{background-color:#fcfcfc;border-bottom:1px solid #fcfcfc;color:#222;font-weight:700}.c-reading-companion__sections-list{list-style:none;padding:0}.c-reading-companion__figures-list,.c-reading-companion__references-list{list-style:none;min-height:389px;padding:0}.c-reading-companion__references-list--numeric{list-style:decimal inside}.c-reading-companion__sections-list{margin:0 0 8px;min-height:50px}.c-reading-companion__section-item{font-size:1rem;padding:0}.c-reading-companion__section-item a{display:block;line-height:1.5;overflow:hidden;padding:8px 0 8px 16px;text-overflow:ellipsis;white-space:nowrap}.c-reading-companion__figure-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:16px 8px 16px 0}.c-reading-companion__figure-item:first-child{border-top:none;padding-top:8px}.c-reading-companion__reference-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:8px 8px 8px 16px}.c-reading-companion__reference-item:first-child{border-top:none}.c-reading-companion__reference-item a{word-break:break-word}.c-reading-companion__reference-citation{display:inline}.c-reading-companion__reference-links{font-size:.813rem;font-weight:700;list-style:none;margin:8px 0 0;padding:0;text-align:right}.c-reading-companion__reference-links>a{display:inline-block;padding-left:8px}.c-reading-companion__reference-links>a:first-child{display:inline-block;padding-left:0}.c-reading-companion__figure-title{display:block;margin:0 0 8px}.c-reading-companion__figure-links{display:flex;justify-content:space-between;margin:8px 0 0}.c-reading-companion__figure-links>a{align-items:center;display:flex}.c-reading-companion__figure-full-link svg{height:.8em;margin-left:2px}.c-reading-companion__panel{border-top:none;display:none;margin-top:0;padding-top:0}.c-reading-companion__panel--active{display:block}.c-pdf-download__link .u-icon{padding-top:2px}.c-pdf-download{display:flex;margin-bottom:16px;max-height:48px}@media only screen and (min-width:540px){.c-pdf-download{max-height:none}}@media only screen and (min-width:1024px){.c-pdf-download{max-height:48px}}.c-pdf-download__link{display:flex;flex:1 1 0%;padding:13px 24px!important}.c-pdf-download__text{padding-right:4px}@media only screen and (max-width:539px){.c-pdf-download__text{text-transform:capitalize}}@media only screen and (min-width:540px){.c-pdf-download__text{padding-right:8px}}.c-pdf-container{display:flex;justify-content:flex-end}@media only screen and (max-width:539px){.c-pdf-container .c-pdf-download{display:flex;flex-basis:100%}}.u-display-none{display:none}.js .u-js-hide,.u-hide{display:none;visibility:hidden}.u-hide:first-child+*{margin-block-start:0}.u-visually-hidden{clip:rect(0,0,0,0);border:0;height:1px;margin:-100%;overflow:hidden;padding:0;position:absolute!important;width:1px}@media print{.u-hide-print{display:none}}@media only screen and (min-width:1024px){.u-hide-at-lg{display:none;visibility:hidden}}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.hide{display:none;visibility:hidden}.c-journal-header__title>a{color:inherit}.c-article-associated-content__container .c-article-associated-content__collection.collection~.c-article-associated-content__collection.collection .c-article-associated-content__collection-label,.c-article-associated-content__container .c-article-associated-content__collection.section~.c-article-associated-content__collection.section .c-article-associated-content__collection-label,.c-article-associated-content__container .c-article-associated-content__title{display:none}.c-article-associated-content__container a{text-decoration:underline}.c-article-associated-content__container .c-article-associated-content__collection.collection .c-article-associated-content__collection-label,.c-article-associated-content__container .c-article-associated-content__collection.section .c-article-associated-content__collection-label{display:block}.c-article-associated-content__container .c-article-associated-content__collection.collection,.c-article-associated-content__container .c-article-associated-content__collection.section{margin-bottom:5px}.c-article-associated-content__container .c-article-associated-content__collection.section~.c-article-associated-content__collection.collection{margin-top:28px}.c-article-associated-content__container .c-article-associated-content__collection:first-child{margin-top:0}.c-article-associated-content__container .c-article-associated-content__collection-label{color:#1b3051;margin-bottom:8px}.c-article-associated-content__container .c-article-associated-content__collection-title{font-size:1.0625rem;font-weight:400} }</style> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/static/app-springeropen/css/enhanced-3013c4b686.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/static/app-springeropen/css/enhanced-article-49340521ae.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <script type="text/javascript"> config = { env: 'live', site: 'advancesincontinuousanddiscretemodels.springeropen.com', siteWithPath: 'advancesincontinuousanddiscretemodels.springeropen.com' + window.location.pathname, twitterHashtag: '', cmsPrefix: 'https://studio-cms.springernature.com/studio/', doi: '10.1186/s13662-016-0916-1', figshareScriptUrl: 'https://widgets.figshare.com/static/figshare.js', hasFigshareInvoked: false, publisherBrand: 'SpringerOpen', mustardcut: false }; </script> <script type="text/javascript" data-test="dataLayer"> window.dataLayer = [{"content":{"article":{"doi":"10.1186/s13662-016-0916-1","articleType":"Research","peerReviewType":"Closed","supplement":null,"keywords":"fractional Brownian motion;existence and uniqueness;stochastic differential equations;non-Lipschitz condition"},"contentInfo":{"imprint":"SpringerOpen","title":"On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion","publishedAt":1469232000000,"publishedAtDate":"2016-07-23","author":["Bin Pei","Yong Xu"],"collection":[]},"attributes":{"deliveryPlatform":"oscar","template":"classic","cms":null,"copyright":{"creativeCommonsType":"CC BY","openAccess":true},"environment":"live"},"journal":{"siteKey":"advancesincontinuousanddiscretemodels.springeropen.com","volume":"2016","issue":"1","title":"Advances in Continuous and Discrete Models","type":null,"journalID":13662,"section":[]},"category":{"pmc":{"primarySubject":"Mathematics"},"contentType":"Research","publishingSegment":"Math-12","snt":["Difference and Functional Equations","Mathematics","Analysis","Functional Analysis","Differential Equations"]}},"session":{"authentication":{"authenticationID":[]}},"version":"1.0.0","page":{"category":{"pageType":"article"},"attributes":{"featureFlags":[],"environment":"live","darwin":false}},"japan":false,"event":"dataLayerCreated","collection":null,"publisherBrand":"SpringerOpen"}]; </script> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ ga4MeasurementId: 'G-PJCTJWPV25', ga360TrackingId: 'UA-54492316-9', twitterId: 'o47a2', baiduId: '29dee5557e2c7961c284a143a770fac0', ga4ServerUrl: 'https://collect.biomedcentral.com', imprint: 'springeropen' }); </script> <script> (function(w, d) { w.config = w.config || {}; w.config.mustardcut = false; if (w.matchMedia && w.matchMedia('only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)').matches) { w.config.mustardcut = true; d.classList.add('js'); d.classList.remove('grade-c'); d.classList.remove('no-js'); } })(window, document.documentElement); </script> <script> (function () { if ( typeof window.CustomEvent === "function" ) return false; function CustomEvent ( event, params ) { params = params || { bubbles: false, cancelable: false, detail: null }; var evt = document.createEvent( 'CustomEvent' ); evt.initCustomEvent( event, params.bubbles, params.cancelable, params.detail ); return evt; } CustomEvent.prototype = window.Event.prototype; window.CustomEvent = CustomEvent; })(); </script> <script class="js-entry"> if (window.config.mustardcut) { (function(w, d) { window.Component = {}; window.suppressShareButton = true; window.onArticlePage = true; var currentScript = d.currentScript || d.head.querySelector('script.js-entry'); function catchNoModuleSupport() { var scriptEl = d.createElement('script'); return (!('noModule' in scriptEl) && 'onbeforeload' in scriptEl) } var headScripts = [ {'src': '/static/js/polyfill-es5-bundle-572d4fec60.js', 'async': false} ]; var bodyScripts = [ {'src': '/static/js/app-es5-bundle-d0ac94c97e.js', 'async': false, 'module': false}, {'src': '/static/js/app-es6-bundle-5ee1a6879c.js', 'async': false, 'module': true} , {'src': '/static/js/global-article-es5-bundle-4d00f8c6b9.js', 'async': false, 'module': false}, {'src': '/static/js/global-article-es6-bundle-31e6854365.js', 'async': false, 'module': true} ]; function createScript(script) { var scriptEl = d.createElement('script'); scriptEl.src = script.src; scriptEl.async = script.async; if (script.module === true) { scriptEl.type = "module"; if (catchNoModuleSupport()) { scriptEl.src = ''; } } else if (script.module === false) { scriptEl.setAttribute('nomodule', true) } if (script.charset) { scriptEl.setAttribute('charset', script.charset); } return scriptEl; } for (var i = 0; i < headScripts.length; ++i) { var scriptEl = createScript(headScripts[i]); currentScript.parentNode.insertBefore(scriptEl, currentScript.nextSibling); } d.addEventListener('DOMContentLoaded', function() { for (var i = 0; i < bodyScripts.length; ++i) { var scriptEl = createScript(bodyScripts[i]); d.body.appendChild(scriptEl); } }); // Webfont repeat view var config = w.config; if (config && config.publisherBrand && sessionStorage.fontsLoaded === 'true') { d.documentElement.className += ' webfonts-loaded'; } })(window, document); } </script> <script data-src="https://cdn.optimizely.com/js/27195530232.js" data-cc-script="C03"></script> <script data-test="gtm-head"> window.initGTM = function() { (function (w, d, s, l, i) { w[l] = w[l] || []; w[l].push({'gtm.start': new Date().getTime(), event: 'gtm.js'}); var f = d.getElementsByTagName(s)[0], j = d.createElement(s), dl = l != 'dataLayer' ? '&l=' + l : ''; j.async = true; j.src = 'https://sgtm.springernature.com/gtm.js?id=' + i + dl; f.parentNode.insertBefore(j, f); })(window, document, 'script', 'dataLayer', 'GTM-MRVXSHQ'); } </script> <meta name="360-site-verification" content="6ebcece7bd3d627674314d9ecc077510" /> <script> (function (w, d, t) { function cc() { var h = w.location.hostname; var e = d.createElement(t), s = d.getElementsByTagName(t)[0]; if (h.indexOf('springer.com') > -1 && h.indexOf('biomedcentral.com') === -1 && h.indexOf('springeropen.com') === -1) { if (h.indexOf('link-qa.springer.com') > -1 || h.indexOf('test-www.springer.com') > -1) { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-54.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-54.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('biomedcentral.com') > -1) { if (h.indexOf('biomedcentral.com.qa') > -1) { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-39.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-39.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springeropen.com') > -1) { if (h.indexOf('springeropen.com.qa') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springernature.com') > -1) { if (h.indexOf('beta-qa.springernature.com') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } } else { e.src = '/static/js/cookie-consent-es5-bundle-cb57c2c98a.js'; e.setAttribute('data-consent', h); } s.insertAdjacentElement('afterend', e); } cc(); })(window, document, 'script'); </script> <link rel="canonical" href="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-016-0916-1"/> <meta property="og:url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-016-0916-1"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="SpringerOpen"/> <meta property="og:title" content="On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion - Advances in Continuous and Discrete Models"/> <meta property="og:description" content="In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter H ∈ ( 1 2 , 1 ) $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/13662"/> <script type="application/ld+json">{"mainEntity":{"headline":"On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion","description":"In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter \n \n \n \n \n \n \n $H\\in(\\frac{1}{2},1)$\n . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study.","datePublished":"2016-07-23T00:00:00Z","dateModified":"2016-07-23T00:00:00Z","pageStart":"1","pageEnd":"15","sameAs":"https://doi.org/10.1186/s13662-016-0916-1","keywords":["fractional Brownian motion","existence and uniqueness","stochastic differential equations","non-Lipschitz condition","Difference and Functional Equations","Mathematics","general","Analysis","Functional Analysis","Ordinary Differential Equations","Partial Differential Equations"],"image":[],"isPartOf":{"name":"Advances in Difference Equations","issn":["1687-1847"],"volumeNumber":"2016","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer International Publishing","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Bin Pei","affiliation":[{"name":"Northwestern Polytechnical University","address":{"name":"Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Yong Xu","affiliation":[{"name":"Northwestern Polytechnical University","address":{"name":"Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"email":"hsux3@nwpu.edu.cn","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="journal journal-fulltext" > <div class="ctm"></div>  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="a" d="M0 .74h56.72v55.24H0z"/></defs><symbol id="icon-access" viewBox="0 0 18 18"><path d="m14 8c.5522847 0 1 .44771525 1 1v7h2.5c.2761424 0 .5.2238576.5.5v1.5h-18v-1.5c0-.2761424.22385763-.5.5-.5h2.5v-7c0-.55228475.44771525-1 1-1s1 .44771525 1 1v6.9996556h8v-6.9996556c0-.55228475.4477153-1 1-1zm-8 0 2 1v5l-2 1zm6 0v7l-2-1v-5zm-2.42653766-7.59857636 7.03554716 4.92488299c.4162533.29137735.5174853.86502537.226108 1.28127873-.1721584.24594054-.4534847.39241464-.7536934.39241464h-14.16284822c-.50810197 0-.92-.41189803-.92-.92 0-.30020869.1464741-.58153499.39241464-.75369337l7.03554714-4.92488299c.34432015-.2410241.80260453-.2410241 1.14692468 0zm-.57346234 2.03988748-3.65526982 2.55868888h7.31053962z" fill-rule="evenodd"/></symbol><symbol id="icon-account" viewBox="0 0 18 18"><path d="m10.2379028 16.9048051c1.3083556-.2032362 2.5118471-.7235183 3.5294683-1.4798399-.8731327-2.5141501-2.0638925-3.935978-3.7673711-4.3188248v-1.27684611c1.1651924-.41183641 2-1.52307546 2-2.82929429 0-1.65685425-1.3431458-3-3-3-1.65685425 0-3 1.34314575-3 3 0 1.30621883.83480763 2.41745788 2 2.82929429v1.27684611c-1.70347856.3828468-2.89423845 1.8046747-3.76737114 4.3188248 1.01762123.7563216 2.22111275 1.2766037 3.52946833 1.4798399.40563808.0629726.81921174.0951949 1.23790281.0951949s.83226473-.0322223 1.2379028-.0951949zm4.3421782-2.1721994c1.4927655-1.4532925 2.419919-3.484675 2.419919-5.7326057 0-4.418278-3.581722-8-8-8s-8 3.581722-8 8c0 2.2479307.92715352 4.2793132 2.41991895 5.7326057.75688473-2.0164459 1.83949951-3.6071894 3.48926591-4.3218837-1.14534283-.70360829-1.90918486-1.96796271-1.90918486-3.410722 0-2.209139 1.790861-4 4-4s4 1.790861 4 4c0 1.44275929-.763842 2.70711371-1.9091849 3.410722 1.6497664.7146943 2.7323812 2.3054378 3.4892659 4.3218837zm-5.580081 3.2673943c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9z" fill-rule="evenodd"/></symbol><symbol id="icon-alert" viewBox="0 0 18 18"><path d="m4 10h2.5c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-3.08578644l-1.12132034 1.1213203c-.18753638.1875364-.29289322.4418903-.29289322.7071068v.1715729h14v-.1715729c0-.2652165-.1053568-.5195704-.2928932-.7071068l-1.7071068-1.7071067v-3.4142136c0-2.76142375-2.2385763-5-5-5-2.76142375 0-5 2.23857625-5 5zm3 4c0 1.1045695.8954305 2 2 2s2-.8954305 2-2zm-5 0c-.55228475 0-1-.4477153-1-1v-.1715729c0-.530433.21071368-1.0391408.58578644-1.4142135l1.41421356-1.4142136v-3c0-3.3137085 2.6862915-6 6-6s6 2.6862915 6 6v3l1.4142136 1.4142136c.3750727.3750727.5857864.8837805.5857864 1.4142135v.1715729c0 .5522847-.4477153 1-1 1h-4c0 1.6568542-1.3431458 3-3 3-1.65685425 0-3-1.3431458-3-3z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-broad" viewBox="0 0 16 16"><path d="m6.10307866 2.97190702v7.69043288l2.44965196-2.44676915c.38776071-.38730439 1.0088052-.39493524 1.38498697-.01919617.38609051.38563612.38643641 1.01053024-.00013864 1.39665039l-4.12239817 4.11754683c-.38616704.3857126-1.01187344.3861062-1.39846576-.0000311l-4.12258206-4.11773056c-.38618426-.38572979-.39254614-1.00476697-.01636437-1.38050605.38609047-.38563611 1.01018509-.38751562 1.4012233.00306241l2.44985644 2.4469734v-8.67638639c0-.54139983.43698413-.98042709.98493125-.98159081l7.89910522-.0043627c.5451687 0 .9871152.44142642.9871152.98595351s-.4419465.98595351-.9871152.98595351z" fill-rule="evenodd" transform="matrix(-1 0 0 -1 14 15)"/></symbol><symbol id="icon-arrow-down" viewBox="0 0 16 16"><path d="m3.28337502 11.5302405 4.03074001 4.176208c.37758093.3912076.98937525.3916069 1.367372-.0000316l4.03091977-4.1763942c.3775978-.3912252.3838182-1.0190815.0160006-1.4001736-.3775061-.39113013-.9877245-.39303641-1.3700683.003106l-2.39538585 2.4818345v-11.6147896l-.00649339-.11662112c-.055753-.49733869-.46370161-.88337888-.95867408-.88337888-.49497246 0-.90292107.38604019-.95867408.88337888l-.00649338.11662112v11.6147896l-2.39518594-2.4816273c-.37913917-.39282218-.98637524-.40056175-1.35419292-.0194697-.37750607.3911302-.37784433 1.0249269.00013556 1.4165479z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-left" viewBox="0 0 16 16"><path d="m4.46975946 3.28337502-4.17620792 4.03074001c-.39120768.37758093-.39160691.98937525.0000316 1.367372l4.1763942 4.03091977c.39122514.3775978 1.01908149.3838182 1.40017357.0160006.39113012-.3775061.3930364-.9877245-.00310603-1.3700683l-2.48183446-2.39538585h11.61478958l.1166211-.00649339c.4973387-.055753.8833789-.46370161.8833789-.95867408 0-.49497246-.3860402-.90292107-.8833789-.95867408l-.1166211-.00649338h-11.61478958l2.4816273-2.39518594c.39282216-.37913917.40056173-.98637524.01946965-1.35419292-.39113012-.37750607-1.02492687-.37784433-1.41654791.00013556z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-right" viewBox="0 0 16 16"><path d="m11.5302405 12.716625 4.176208-4.03074003c.3912076-.37758093.3916069-.98937525-.0000316-1.367372l-4.1763942-4.03091981c-.3912252-.37759778-1.0190815-.38381821-1.4001736-.01600053-.39113013.37750607-.39303641.98772445.003106 1.37006824l2.4818345 2.39538588h-11.6147896l-.11662112.00649339c-.49733869.055753-.88337888.46370161-.88337888.95867408 0 .49497246.38604019.90292107.88337888.95867408l.11662112.00649338h11.6147896l-2.4816273 2.39518592c-.39282218.3791392-.40056175.9863753-.0194697 1.3541929.3911302.3775061 1.0249269.3778444 1.4165479-.0001355z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-sub" viewBox="0 0 16 16"><path d="m7.89692134 4.97190702v7.69043288l-2.44965196-2.4467692c-.38776071-.38730434-1.0088052-.39493519-1.38498697-.0191961-.38609047.3856361-.38643643 1.0105302.00013864 1.3966504l4.12239817 4.1175468c.38616704.3857126 1.01187344.3861062 1.39846576-.0000311l4.12258202-4.1177306c.3861843-.3857298.3925462-1.0047669.0163644-1.380506-.3860905-.38563612-1.0101851-.38751563-1.4012233.0030624l-2.44985643 2.4469734v-8.67638639c0-.54139983-.43698413-.98042709-.98493125-.98159081l-7.89910525-.0043627c-.54516866 0-.98711517.44142642-.98711517.98595351s.44194651.98595351.98711517.98595351z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-up" viewBox="0 0 16 16"><path d="m12.716625 4.46975946-4.03074003-4.17620792c-.37758093-.39120768-.98937525-.39160691-1.367372.0000316l-4.03091981 4.1763942c-.37759778.39122514-.38381821 1.01908149-.01600053 1.40017357.37750607.39113012.98772445.3930364 1.37006824-.00310603l2.39538588-2.48183446v11.61478958l.00649339.1166211c.055753.4973387.46370161.8833789.95867408.8833789.49497246 0 .90292107-.3860402.95867408-.8833789l.00649338-.1166211v-11.61478958l2.39518592 2.4816273c.3791392.39282216.9863753.40056173 1.3541929.01946965.3775061-.39113012.3778444-1.02492687-.0001355-1.41654791z" fill-rule="evenodd"/></symbol><symbol id="icon-article" viewBox="0 0 18 18"><path d="m13 15v-12.9906311c0-.0073595-.0019884-.0093689.0014977-.0093689l-11.00158888.00087166v13.00506804c0 .5482678.44615281.9940603.99415146.9940603h10.27350412c-.1701701-.2941734-.2675644-.6357129-.2675644-1zm-12 .0059397v-13.00506804c0-.5562408.44704472-1.00087166.99850233-1.00087166h11.00299537c.5510129 0 .9985023.45190985.9985023 1.0093689v2.9906311h3v9.9914698c0 1.1065798-.8927712 2.0085302-1.9940603 2.0085302h-12.01187942c-1.09954652 0-1.99406028-.8927712-1.99406028-1.9940603zm13-9.0059397v9c0 .5522847.4477153 1 1 1s1-.4477153 1-1v-9zm-10-2h7v4h-7zm1 1v2h5v-2zm-1 4h7v1h-7zm0 2h7v1h-7zm0 2h7v1h-7z" fill-rule="evenodd"/></symbol><symbol id="icon-audio" viewBox="0 0 18 18"><path d="m13.0957477 13.5588459c-.195279.1937043-.5119137.193729-.7072234.0000551-.1953098-.193674-.1953346-.5077061-.0000556-.7014104 1.0251004-1.0168342 1.6108711-2.3905226 1.6108711-3.85745208 0-1.46604976-.5850634-2.83898246-1.6090736-3.85566829-.1951894-.19379323-.1950192-.50782531.0003802-.70141028.1953993-.19358497.512034-.19341614.7072234.00037709 1.2094886 1.20083761 1.901635 2.8250555 1.901635 4.55670148 0 1.73268608-.6929822 3.35779608-1.9037571 4.55880738zm2.1233994 2.1025159c-.195234.193749-.5118687.1938462-.7072235.0002171-.1953548-.1936292-.1954528-.5076613-.0002189-.7014104 1.5832215-1.5711805 2.4881302-3.6939808 2.4881302-5.96012998 0-2.26581266-.9046382-4.3883241-2.487443-5.95944795-.1952117-.19377107-.1950777-.50780316.0002993-.70141031s.5120117-.19347426.7072234.00029682c1.7683321 1.75528196 2.7800854 4.12911258 2.7800854 6.66056144 0 2.53182498-1.0120556 4.90597838-2.7808529 6.66132328zm-14.21898205-3.6854911c-.5523759 0-1.00016505-.4441085-1.00016505-.991944v-3.96777631c0-.54783558.44778915-.99194407 1.00016505-.99194407h2.0003301l5.41965617-3.8393633c.44948677-.31842296 1.07413994-.21516983 1.39520191.23062232.12116339.16823446.18629727.36981184.18629727.57655577v12.01603479c0 .5478356-.44778914.9919441-1.00016505.9919441-.20845738 0-.41170538-.0645985-.58133413-.184766l-5.41965617-3.8393633zm0-.991944h2.32084805l5.68047235 4.0241292v-12.01603479l-5.68047235 4.02412928h-2.32084805z" fill-rule="evenodd"/></symbol><symbol id="icon-block" viewBox="0 0 24 24"><path d="m0 0h24v24h-24z" fill-rule="evenodd"/></symbol><symbol id="icon-book" viewBox="0 0 18 18"><path d="m4 13v-11h1v11h11v-11h-13c-.55228475 0-1 .44771525-1 1v10.2675644c.29417337-.1701701.63571286-.2675644 1-.2675644zm12 1h-13c-.55228475 0-1 .4477153-1 1s.44771525 1 1 1h13zm0 3h-13c-1.1045695 0-2-.8954305-2-2v-12c0-1.1045695.8954305-2 2-2h13c.5522847 0 1 .44771525 1 1v14c0 .5522847-.4477153 1-1 1zm-8.5-13h6c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-6c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm1 2h4c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-4c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-broad" viewBox="0 0 24 24"><path d="m9.18274226 7.81v7.7999954l2.48162734-2.4816273c.3928221-.3928221 1.0219731-.4005617 1.4030652-.0194696.3911301.3911301.3914806 1.0249268-.0001404 1.4165479l-4.17620796 4.1762079c-.39120769.3912077-1.02508144.3916069-1.41671995-.0000316l-4.1763942-4.1763942c-.39122514-.3912251-.39767006-1.0190815-.01657798-1.4001736.39113012-.3911301 1.02337106-.3930364 1.41951349.0031061l2.48183446 2.4818344v-8.7999954c0-.54911294.4426881-.99439484.99778758-.99557515l8.00221246-.00442485c.5522847 0 1 .44771525 1 1s-.4477153 1-1 1z" fill-rule="evenodd" transform="matrix(-1 0 0 -1 20.182742 24.805206)"/></symbol><symbol id="icon-calendar" viewBox="0 0 18 18"><path d="m12.5 0c.2761424 0 .5.21505737.5.49047852v.50952148h2c1.1072288 0 2 .89451376 2 2v12c0 1.1072288-.8945138 2-2 2h-12c-1.1072288 0-2-.8945138-2-2v-12c0-1.1072288.89451376-2 2-2h1v1h-1c-.55393837 0-1 .44579254-1 1v3h14v-3c0-.55393837-.4457925-1-1-1h-2v1.50952148c0 .27088381-.2319336.49047852-.5.49047852-.2761424 0-.5-.21505737-.5-.49047852v-3.01904296c0-.27088381.2319336-.49047852.5-.49047852zm3.5 7h-14v8c0 .5539384.44579254 1 1 1h12c.5539384 0 1-.4457925 1-1zm-11 6v1h-1v-1zm3 0v1h-1v-1zm3 0v1h-1v-1zm-6-2v1h-1v-1zm3 0v1h-1v-1zm6 0v1h-1v-1zm-3 0v1h-1v-1zm-3-2v1h-1v-1zm6 0v1h-1v-1zm-3 0v1h-1v-1zm-5.5-9c.27614237 0 .5.21505737.5.49047852v.50952148h5v1h-5v1.50952148c0 .27088381-.23193359.49047852-.5.49047852-.27614237 0-.5-.21505737-.5-.49047852v-3.01904296c0-.27088381.23193359-.49047852.5-.49047852z" fill-rule="evenodd"/></symbol><symbol id="icon-cart" viewBox="0 0 18 18"><path d="m5 14c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm10 0c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm-10 1c-.55228475 0-1 .4477153-1 1s.44771525 1 1 1 1-.4477153 1-1-.44771525-1-1-1zm10 0c-.5522847 0-1 .4477153-1 1s.4477153 1 1 1 1-.4477153 1-1-.4477153-1-1-1zm-12.82032249-15c.47691417 0 .88746157.33678127.98070211.80449199l.23823144 1.19501025 13.36277974.00045554c.5522847.00001882.9999659.44774934.9999659 1.00004222 0 .07084994-.0075361.14150708-.022474.2107727l-1.2908094 5.98534344c-.1007861.46742419-.5432548.80388386-1.0571651.80388386h-10.24805106c-.59173366 0-1.07142857.4477153-1.07142857 1 0 .5128358.41361449.9355072.94647737.9932723l.1249512.0067277h10.35933776c.2749512 0 .4979349.2228539.4979349.4978051 0 .2749417-.2227336.4978951-.4976753.4980063l-10.35959736.0041886c-1.18346732 0-2.14285714-.8954305-2.14285714-2 0-.6625717.34520317-1.24989198.87690425-1.61383592l-1.63768102-8.19004794c-.01312273-.06561364-.01950005-.131011-.0196107-.19547395l-1.71961253-.00064219c-.27614237 0-.5-.22385762-.5-.5 0-.27614237.22385763-.5.5-.5zm14.53193359 2.99950224h-13.11300004l1.20580469 6.02530174c.11024034-.0163252.22327998-.02480398.33844139-.02480398h10.27064786z"/></symbol><symbol id="icon-chevron-less" viewBox="0 0 10 10"><path d="m5.58578644 4-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4c-.39052429.39052429-1.02368927.39052429-1.41421356 0s-.39052429-1.02368927 0-1.41421356z" fill-rule="evenodd" transform="matrix(0 -1 -1 0 9 9)"/></symbol><symbol id="icon-chevron-more" viewBox="0 0 10 10"><path d="m5.58578644 6-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4.00000002c-.39052429.3905243-1.02368927.3905243-1.41421356 0s-.39052429-1.02368929 0-1.41421358z" fill-rule="evenodd" transform="matrix(0 1 -1 0 11 1)"/></symbol><symbol id="icon-chevron-right" viewBox="0 0 10 10"><path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/></symbol><symbol id="icon-circle-fill" viewBox="0 0 16 16"><path d="m8 14c-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6 6 2.6862915 6 6-2.6862915 6-6 6z" fill-rule="evenodd"/></symbol><symbol id="icon-circle" viewBox="0 0 16 16"><path d="m8 12c2.209139 0 4-1.790861 4-4s-1.790861-4-4-4-4 1.790861-4 4 1.790861 4 4 4zm0 2c-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6 6 2.6862915 6 6-2.6862915 6-6 6z" fill-rule="evenodd"/></symbol><symbol id="icon-citation" viewBox="0 0 18 18"><path d="m8.63593473 5.99995183c2.20913897 0 3.99999997 1.79084375 3.99999997 3.99996146 0 1.40730761-.7267788 2.64486871-1.8254829 3.35783281 1.6240224.6764218 2.8754442 2.0093871 3.4610603 3.6412466l-1.0763845.000006c-.5310008-1.2078237-1.5108121-2.1940153-2.7691712-2.7181346l-.79002167-.329052v-1.023992l.63016577-.4089232c.8482885-.5504661 1.3698342-1.4895187 1.3698342-2.51898361 0-1.65683828-1.3431457-2.99996146-2.99999997-2.99996146-1.65685425 0-3 1.34312318-3 2.99996146 0 1.02946491.52154569 1.96851751 1.36983419 2.51898361l.63016581.4089232v1.023992l-.79002171.329052c-1.25835905.5241193-2.23817037 1.5103109-2.76917113 2.7181346l-1.07638453-.000006c.58561612-1.6318595 1.8370379-2.9648248 3.46106024-3.6412466-1.09870405-.7129641-1.82548287-1.9505252-1.82548287-3.35783281 0-2.20911771 1.790861-3.99996146 4-3.99996146zm7.36897597-4.99995183c1.1018574 0 1.9950893.89353404 1.9950893 2.00274083v5.994422c0 1.10608317-.8926228 2.00274087-1.9950893 2.00274087l-3.0049107-.0009037v-1l3.0049107.00091329c.5490631 0 .9950893-.44783123.9950893-1.00275046v-5.994422c0-.55646537-.4450595-1.00275046-.9950893-1.00275046h-14.00982141c-.54906309 0-.99508929.44783123-.99508929 1.00275046v5.9971821c0 .66666024.33333333.99999036 1 .99999036l2-.00091329v1l-2 .0009037c-1 0-2-.99999041-2-1.99998077v-5.9971821c0-1.10608322.8926228-2.00274083 1.99508929-2.00274083zm-8.5049107 2.9999711c.27614237 0 .5.22385547.5.5 0 .2761349-.22385763.5-.5.5h-4c-.27614237 0-.5-.2238651-.5-.5 0-.27614453.22385763-.5.5-.5zm3 0c.2761424 0 .5.22385547.5.5 0 .2761349-.2238576.5-.5.5h-1c-.27614237 0-.5-.2238651-.5-.5 0-.27614453.22385763-.5.5-.5zm4 0c.2761424 0 .5.22385547.5.5 0 .2761349-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238651-.5-.5 0-.27614453.2238576-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-close" viewBox="0 0 16 16"><path d="m2.29679575 12.2772478c-.39658757.3965876-.39438847 1.0328109-.00062148 1.4265779.39651227.3965123 1.03246768.3934888 1.42657791-.0006214l4.27724782-4.27724787 4.2772478 4.27724787c.3965876.3965875 1.0328109.3943884 1.4265779.0006214.3965123-.3965122.3934888-1.0324677-.0006214-1.4265779l-4.27724787-4.2772478 4.27724787-4.27724782c.3965875-.39658757.3943884-1.03281091.0006214-1.42657791-.3965122-.39651226-1.0324677-.39348875-1.4265779.00062148l-4.2772478 4.27724782-4.27724782-4.27724782c-.39658757-.39658757-1.03281091-.39438847-1.42657791-.00062148-.39651226.39651227-.39348875 1.03246768.00062148 1.42657791l4.27724782 4.27724782z" fill-rule="evenodd"/></symbol><symbol id="icon-collections" viewBox="0 0 18 18"><path d="m15 4c1.1045695 0 2 .8954305 2 2v9c0 1.1045695-.8954305 2-2 2h-8c-1.1045695 0-2-.8954305-2-2h1c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h8c.5128358 0 .9355072-.3860402.9932723-.8833789l.0067277-.1166211v-9c0-.51283584-.3860402-.93550716-.8833789-.99327227l-.1166211-.00672773h-1v-1zm-4-3c1.1045695 0 2 .8954305 2 2v9c0 1.1045695-.8954305 2-2 2h-8c-1.1045695 0-2-.8954305-2-2v-9c0-1.1045695.8954305-2 2-2zm0 1h-8c-.51283584 0-.93550716.38604019-.99327227.88337887l-.00672773.11662113v9c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h8c.5128358 0 .9355072-.3860402.9932723-.8833789l.0067277-.1166211v-9c0-.51283584-.3860402-.93550716-.8833789-.99327227zm-1.5 7c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm0-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm0-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-compare" viewBox="0 0 18 18"><path d="m12 3c3.3137085 0 6 2.6862915 6 6s-2.6862915 6-6 6c-1.0928452 0-2.11744941-.2921742-2.99996061-.8026704-.88181407.5102749-1.90678042.8026704-3.00003939.8026704-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6c1.09325897 0 2.11822532.29239547 3.00096303.80325037.88158756-.51107621 1.90619177-.80325037 2.99903697-.80325037zm-6 1c-2.76142375 0-5 2.23857625-5 5 0 2.7614237 2.23857625 5 5 5 .74397391 0 1.44999672-.162488 2.08451611-.4539116-1.27652344-1.1000812-2.08451611-2.7287264-2.08451611-4.5460884s.80799267-3.44600721 2.08434391-4.5463015c-.63434719-.29121054-1.34037-.4536985-2.08434391-.4536985zm6 0c-.7439739 0-1.4499967.16248796-2.08451611.45391156 1.27652341 1.10008123 2.08451611 2.72872644 2.08451611 4.54608844s-.8079927 3.4460072-2.08434391 4.5463015c.63434721.2912105 1.34037001.4536985 2.08434391.4536985 2.7614237 0 5-2.2385763 5-5 0-2.76142375-2.2385763-5-5-5zm-1.4162763 7.0005324h-3.16744736c.15614659.3572676.35283837.6927622.58425872 1.0006671h1.99892988c.23142036-.3079049.42811216-.6433995.58425876-1.0006671zm.4162763-2.0005324h-4c0 .34288501.0345146.67770871.10025909 1.0011864h3.79948181c.0657445-.32347769.1002591-.65830139.1002591-1.0011864zm-.4158423-1.99953894h-3.16831543c-.13859957.31730812-.24521946.651783-.31578599.99935097h3.79988742c-.0705665-.34756797-.1771864-.68204285-.315786-.99935097zm-1.58295822-1.999926-.08316107.06199199c-.34550042.27081213-.65446126.58611297-.91825862.93727862h2.00044041c-.28418626-.37830727-.6207872-.71499149-.99902072-.99927061z" fill-rule="evenodd"/></symbol><symbol id="icon-download-file" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm0 1h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v14.00982141c0 .5500396.44491393.9950893.99406028.9950893h12.01187942c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717zm-1.5046024 4c.27614237 0 .5.21637201.5.49209595v6.14827645l1.7462789-1.77990922c.1933927-.1971171.5125222-.19455839.7001689-.0069117.1932998.19329992.1910058.50899492-.0027774.70277812l-2.59089271 2.5908927c-.19483374.1948337-.51177825.1937771-.70556873-.0000133l-2.59099079-2.5909908c-.19484111-.1948411-.19043735-.5151448-.00279066-.70279146.19329987-.19329987.50465175-.19237083.70018565.00692852l1.74638684 1.78001764v-6.14827695c0-.27177709.23193359-.49209595.5-.49209595z" fill-rule="evenodd"/></symbol><symbol id="icon-download" viewBox="0 0 16 16"><path d="m12.9975267 12.999368c.5467123 0 1.0024733.4478567 1.0024733 1.000316 0 .5563109-.4488226 1.000316-1.0024733 1.000316h-9.99505341c-.54671233 0-1.00247329-.4478567-1.00247329-1.000316 0-.5563109.44882258-1.000316 1.00247329-1.000316zm-4.9975267-11.999368c.55228475 0 1 .44497754 1 .99589209v6.80214418l2.4816273-2.48241149c.3928222-.39294628 1.0219732-.4006883 1.4030652-.01947579.3911302.39125371.3914806 1.02525073-.0001404 1.41699553l-4.17620792 4.17752758c-.39120769.3913313-1.02508144.3917306-1.41671995-.0000316l-4.17639421-4.17771394c-.39122513-.39134876-.39767006-1.01940351-.01657797-1.40061601.39113012-.39125372 1.02337105-.3931606 1.41951349.00310701l2.48183446 2.48261871v-6.80214418c0-.55001601.44386482-.99589209 1-.99589209z" fill-rule="evenodd"/></symbol><symbol id="icon-editors" viewBox="0 0 18 18"><path d="m8.72592184 2.54588137c-.48811714-.34391207-1.08343326-.54588137-1.72592184-.54588137-1.65685425 0-3 1.34314575-3 3 0 1.02947485.5215457 1.96853646 1.3698342 2.51900785l.6301658.40892721v1.02400182l-.79002171.32905522c-1.93395773.8055207-3.20997829 2.7024791-3.20997829 4.8180274v.9009805h-1v-.9009805c0-2.5479714 1.54557359-4.79153984 3.82548288-5.7411543-1.09870406-.71297106-1.82548288-1.95054399-1.82548288-3.3578652 0-2.209139 1.790861-4 4-4 1.09079823 0 2.07961816.43662103 2.80122451 1.1446278-.37707584.09278571-.7373238.22835063-1.07530267.40125357zm-2.72592184 14.45411863h-1v-.9009805c0-2.5479714 1.54557359-4.7915398 3.82548288-5.7411543-1.09870406-.71297106-1.82548288-1.95054399-1.82548288-3.3578652 0-2.209139 1.790861-4 4-4s4 1.790861 4 4c0 1.40732121-.7267788 2.64489414-1.8254829 3.3578652 2.2799093.9496145 3.8254829 3.1931829 3.8254829 5.7411543v.9009805h-1v-.9009805c0-2.1155483-1.2760206-4.0125067-3.2099783-4.8180274l-.7900217-.3290552v-1.02400184l.6301658-.40892721c.8482885-.55047139 1.3698342-1.489533 1.3698342-2.51900785 0-1.65685425-1.3431458-3-3-3-1.65685425 0-3 1.34314575-3 3 0 1.02947485.5215457 1.96853646 1.3698342 2.51900785l.6301658.40892721v1.02400184l-.79002171.3290552c-1.93395773.8055207-3.20997829 2.7024791-3.20997829 4.8180274z" fill-rule="evenodd"/></symbol><symbol id="icon-email" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587h-14.00982141c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm0 1h-14.00982141c-.54871518 0-.99508929.44887827-.99508929 1.00585866v9.98828264c0 .5572961.44630695 1.0058587.99508929 1.0058587h14.00982141c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-.0049107 2.55749512v1.44250488l-7 4-7-4v-1.44250488l7 4z" fill-rule="evenodd"/></symbol><symbol id="icon-error" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm2.8630343 4.71100931-2.8630343 2.86303426-2.86303426-2.86303426c-.39658757-.39658757-1.03281091-.39438847-1.4265779-.00062147-.39651227.39651226-.39348876 1.03246767.00062147 1.4265779l2.86303426 2.86303426-2.86303426 2.8630343c-.39658757.3965875-.39438847 1.0328109-.00062147 1.4265779.39651226.3965122 1.03246767.3934887 1.4265779-.0006215l2.86303426-2.8630343 2.8630343 2.8630343c.3965875.3965876 1.0328109.3943885 1.4265779.0006215.3965122-.3965123.3934887-1.0324677-.0006215-1.4265779l-2.8630343-2.8630343 2.8630343-2.86303426c.3965876-.39658757.3943885-1.03281091.0006215-1.4265779-.3965123-.39651227-1.0324677-.39348876-1.4265779.00062147z" fill-rule="evenodd"/></symbol><symbol id="icon-ethics" viewBox="0 0 18 18"><path d="m6.76384967 1.41421356.83301651-.8330165c.77492941-.77492941 2.03133823-.77492941 2.80626762 0l.8330165.8330165c.3750728.37507276.8837806.58578644 1.4142136.58578644h1.3496361c1.1045695 0 2 .8954305 2 2v1.34963611c0 .53043298.2107137 1.03914081.5857864 1.41421356l.8330165.83301651c.7749295.77492941.7749295 2.03133823 0 2.80626762l-.8330165.8330165c-.3750727.3750728-.5857864.8837806-.5857864 1.4142136v1.3496361c0 1.1045695-.8954305 2-2 2h-1.3496361c-.530433 0-1.0391408.2107137-1.4142136.5857864l-.8330165.8330165c-.77492939.7749295-2.03133821.7749295-2.80626762 0l-.83301651-.8330165c-.37507275-.3750727-.88378058-.5857864-1.41421356-.5857864h-1.34963611c-1.1045695 0-2-.8954305-2-2v-1.3496361c0-.530433-.21071368-1.0391408-.58578644-1.4142136l-.8330165-.8330165c-.77492941-.77492939-.77492941-2.03133821 0-2.80626762l.8330165-.83301651c.37507276-.37507275.58578644-.88378058.58578644-1.41421356v-1.34963611c0-1.1045695.8954305-2 2-2h1.34963611c.53043298 0 1.03914081-.21071368 1.41421356-.58578644zm-1.41421356 1.58578644h-1.34963611c-.55228475 0-1 .44771525-1 1v1.34963611c0 .79564947-.31607052 1.55871121-.87867966 2.12132034l-.8330165.83301651c-.38440512.38440512-.38440512 1.00764896 0 1.39205408l.8330165.83301646c.56260914.5626092.87867966 1.3256709.87867966 2.1213204v1.3496361c0 .5522847.44771525 1 1 1h1.34963611c.79564947 0 1.55871121.3160705 2.12132034.8786797l.83301651.8330165c.38440512.3844051 1.00764896.3844051 1.39205408 0l.83301646-.8330165c.5626092-.5626092 1.3256709-.8786797 2.1213204-.8786797h1.3496361c.5522847 0 1-.4477153 1-1v-1.3496361c0-.7956495.3160705-1.5587112.8786797-2.1213204l.8330165-.83301646c.3844051-.38440512.3844051-1.00764896 0-1.39205408l-.8330165-.83301651c-.5626092-.56260913-.8786797-1.32567087-.8786797-2.12132034v-1.34963611c0-.55228475-.4477153-1-1-1h-1.3496361c-.7956495 0-1.5587112-.31607052-2.1213204-.87867966l-.83301646-.8330165c-.38440512-.38440512-1.00764896-.38440512-1.39205408 0l-.83301651.8330165c-.56260913.56260914-1.32567087.87867966-2.12132034.87867966zm3.58698944 11.4960218c-.02081224.002155-.04199226.0030286-.06345763.002542-.98766446-.0223875-1.93408568-.3063547-2.75885125-.8155622-.23496767-.1450683-.30784554-.4531483-.16277726-.688116.14506827-.2349677.45314827-.3078455.68811595-.1627773.67447084.4164161 1.44758575.6483839 2.25617384.6667123.01759529.0003988.03495764.0017019.05204365.0038639.01713363-.0017748.03452416-.0026845.05212715-.0026845 2.4852814 0 4.5-2.0147186 4.5-4.5 0-1.04888973-.3593547-2.04134635-1.0074477-2.83787157-.1742817-.21419731-.1419238-.5291218.0722736-.70340353.2141973-.17428173.5291218-.14192375.7034035.07227357.7919032.97327203 1.2317706 2.18808682 1.2317706 3.46900153 0 3.0375661-2.4624339 5.5-5.5 5.5-.02146768 0-.04261937-.0013529-.06337445-.0039782zm1.57975095-10.78419583c.2654788.07599731.419084.35281842.3430867.61829728-.0759973.26547885-.3528185.419084-.6182973.3430867-.37560116-.10752146-.76586237-.16587951-1.15568824-.17249193-2.5587807-.00064534-4.58547766 2.00216524-4.58547766 4.49928198 0 .62691557.12797645 1.23496.37274865 1.7964426.11035133.2531347-.0053975.5477984-.25853224.6581497-.25313473.1103514-.54779841-.0053975-.65814974-.2585322-.29947131-.6869568-.45606667-1.43097603-.45606667-2.1960601 0-3.05211432 2.47714695-5.50006595 5.59399617-5.49921198.48576182.00815502.96289603.0795037 1.42238033.21103795zm-1.9766658 6.41091303 2.69835-2.94655317c.1788432-.21040373.4943901-.23598862.7047939-.05714545.2104037.17884318.2359886.49439014.0571454.70479387l-3.01637681 3.34277395c-.18039088.1999106-.48669547.2210637-.69285412.0478478l-1.93095347-1.62240047c-.21213845-.17678204-.24080048-.49206439-.06401844-.70420284.17678204-.21213844.49206439-.24080048.70420284-.06401844z" fill-rule="evenodd"/></symbol><symbol id="icon-expand"><path d="M7.498 11.918a.997.997 0 0 0-.003-1.411.995.995 0 0 0-1.412-.003l-4.102 4.102v-3.51A1 1 0 0 0 .98 10.09.992.992 0 0 0 0 11.092V17c0 .554.448 1.002 1.002 1.002h5.907c.554 0 1.002-.45 1.002-1.003 0-.539-.45-.978-1.006-.978h-3.51zm3.005-5.835a.997.997 0 0 0 .003 1.412.995.995 0 0 0 1.411.003l4.103-4.103v3.51a1 1 0 0 0 1.001 1.006A.992.992 0 0 0 18 6.91V1.002A1 1 0 0 0 17 0h-5.907a1.003 1.003 0 0 0-1.002 1.003c0 .539.45.978 1.006.978h3.51z" fill-rule="evenodd"/></symbol><symbol id="icon-explore" viewBox="0 0 18 18"><path d="m9 17c4.418278 0 8-3.581722 8-8s-3.581722-8-8-8-8 3.581722-8 8 3.581722 8 8 8zm0 1c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9zm0-2.5c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5c2.969509 0 5.400504-2.3575119 5.497023-5.31714844.0090007-.27599565.2400359-.49243782.5160315-.48343711.2759957.0090007.4924378.2400359.4834371.51603155-.114093 3.4985237-2.9869632 6.284554-6.4964916 6.284554zm-.29090657-12.99359748c.27587424-.01216621.50937715.20161139.52154336.47748563.01216621.27587423-.20161139.50937715-.47748563.52154336-2.93195733.12930094-5.25315116 2.54886451-5.25315116 5.49456849 0 .27614237-.22385763.5-.5.5s-.5-.22385763-.5-.5c0-3.48142406 2.74307146-6.34074398 6.20909343-6.49359748zm1.13784138 8.04763908-1.2004882-1.20048821c-.19526215-.19526215-.19526215-.51184463 0-.70710678s.51184463-.19526215.70710678 0l1.20048821 1.2004882 1.6006509-4.00162734-4.50670359 1.80268144-1.80268144 4.50670359zm4.10281269-6.50378907-2.6692597 6.67314927c-.1016411.2541026-.3029834.4554449-.557086.557086l-6.67314927 2.6692597 2.66925969-6.67314926c.10164107-.25410266.30298336-.45544495.55708602-.55708602z" fill-rule="evenodd"/></symbol><symbol id="icon-filter" viewBox="0 0 16 16"><path d="m14.9738641 0c.5667192 0 1.0261359.4477136 1.0261359 1 0 .24221858-.0902161.47620768-.2538899.65849851l-5.6938314 6.34147206v5.49997973c0 .3147562-.1520673.6111434-.4104543.7999971l-2.05227171 1.4999945c-.45337535.3313696-1.09655869.2418269-1.4365902-.1999993-.13321514-.1730955-.20522717-.3836284-.20522717-.5999978v-6.99997423l-5.69383133-6.34147206c-.3731872-.41563511-.32996891-1.0473954.09653074-1.41107611.18705584-.15950448.42716133-.2474224.67571519-.2474224zm-5.9218641 8.5h-2.105v6.491l.01238459.0070843.02053271.0015705.01955278-.0070558 2.0532976-1.4990996zm-8.02585008-7.5-.01564945.00240169 5.83249953 6.49759831h2.313l5.836-6.499z"/></symbol><symbol id="icon-home" viewBox="0 0 18 18"><path d="m9 5-6 6v5h4v-4h4v4h4v-5zm7 6.5857864v4.4142136c0 .5522847-.4477153 1-1 1h-5v-4h-2v4h-5c-.55228475 0-1-.4477153-1-1v-4.4142136c-.25592232 0-.51184464-.097631-.70710678-.2928932l-.58578644-.5857864c-.39052429-.3905243-.39052429-1.02368929 0-1.41421358l8.29289322-8.29289322 8.2928932 8.29289322c.3905243.39052429.3905243 1.02368928 0 1.41421358l-.5857864.5857864c-.1952622.1952622-.4511845.2928932-.7071068.2928932zm-7-9.17157284-7.58578644 7.58578644.58578644.5857864 7-6.99999996 7 6.99999996.5857864-.5857864z" fill-rule="evenodd"/></symbol><symbol id="icon-image" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm-3.49645283 10.1752453-3.89407257 6.7495552c.11705545.048464.24538859.0751995.37998328.0751995h10.60290092l-2.4329715-4.2154691-1.57494129 2.7288098zm8.49779013 6.8247547c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v13.98991071l4.50814957-7.81026689 3.08089884 5.33809539 1.57494129-2.7288097 3.5875735 6.2159812zm-3.0059397-11c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm0 1c-.5522847 0-1 .44771525-1 1s.4477153 1 1 1 1-.44771525 1-1-.4477153-1-1-1z" fill-rule="evenodd"/></symbol><symbol id="icon-info" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm0 7h-1.5l-.11662113.00672773c-.49733868.05776511-.88337887.48043643-.88337887.99327227 0 .47338693.32893365.86994729.77070917.97358929l.1126697.01968298.11662113.00672773h.5v3h-.5l-.11662113.0067277c-.42082504.0488782-.76196299.3590206-.85696816.7639815l-.01968298.1126697-.00672773.1166211.00672773.1166211c.04887817.4208251.35902055.761963.76398144.8569682l.1126697.019683.11662113.0067277h3l.1166211-.0067277c.4973387-.0577651.8833789-.4804365.8833789-.9932723 0-.4733869-.3289337-.8699473-.7707092-.9735893l-.1126697-.019683-.1166211-.0067277h-.5v-4l-.00672773-.11662113c-.04887817-.42082504-.35902055-.76196299-.76398144-.85696816l-.1126697-.01968298zm0-3.25c-.69035594 0-1.25.55964406-1.25 1.25s.55964406 1.25 1.25 1.25 1.25-.55964406 1.25-1.25-.55964406-1.25-1.25-1.25z" fill-rule="evenodd"/></symbol><symbol id="icon-institution" viewBox="0 0 18 18"><path d="m7 16.9998189v-2.0003623h4v2.0003623h2v-3.0005434h-8v3.0005434zm-3-10.00181122h-1.52632364c-.27614237 0-.5-.22389817-.5-.50009056 0-.13995446.05863589-.27350497.16166338-.36820841l1.23156713-1.13206327h-2.36690687v12.00217346h3v-2.0003623h-3v-1.0001811h3v-1.0001811h1v-4.00072448h-1zm10 0v2.00036224h-1v4.00072448h1v1.0001811h3v1.0001811h-3v2.0003623h3v-12.00217346h-2.3695309l1.2315671 1.13206327c.2033191.186892.2166633.50325042.0298051.70660631-.0946863.10304615-.2282126.16169266-.3681417.16169266zm3-3.00054336c.5522847 0 1 .44779634 1 1.00018112v13.00235456h-18v-13.00235456c0-.55238478.44771525-1.00018112 1-1.00018112h3.45499992l4.20535144-3.86558216c.19129876-.17584288.48537447-.17584288.67667324 0l4.2053514 3.86558216zm-4 3.00054336h-8v1.00018112h8zm-2 6.00108672h1v-4.00072448h-1zm-1 0v-4.00072448h-2v4.00072448zm-3 0v-4.00072448h-1v4.00072448zm8-4.00072448c.5522847 0 1 .44779634 1 1.00018112v2.00036226h-2v-2.00036226c0-.55238478.4477153-1.00018112 1-1.00018112zm-12 0c.55228475 0 1 .44779634 1 1.00018112v2.00036226h-2v-2.00036226c0-.55238478.44771525-1.00018112 1-1.00018112zm5.99868798-7.81907007-5.24205601 4.81852671h10.48411203zm.00131202 3.81834559c-.55228475 0-1-.44779634-1-1.00018112s.44771525-1.00018112 1-1.00018112 1 .44779634 1 1.00018112-.44771525 1.00018112-1 1.00018112zm-1 11.00199236v1.0001811h2v-1.0001811z" fill-rule="evenodd"/></symbol><symbol id="icon-location" viewBox="0 0 18 18"><path d="m9.39521328 16.2688008c.79596342-.7770119 1.59208152-1.6299956 2.33285652-2.5295081 1.4020032-1.7024324 2.4323601-3.3624519 2.9354918-4.871847.2228715-.66861448.3364384-1.29323246.3364384-1.8674457 0-3.3137085-2.6862915-6-6-6-3.36356866 0-6 2.60156856-6 6 0 .57421324.11356691 1.19883122.3364384 1.8674457.50313169 1.5093951 1.53348863 3.1694146 2.93549184 4.871847.74077492.8995125 1.53689309 1.7524962 2.33285648 2.5295081.13694479.1336842.26895677.2602648.39521328.3793207.12625651-.1190559.25826849-.2456365.39521328-.3793207zm-.39521328 1.7311992s-7-6-7-11c0-4 3.13400675-7 7-7 3.8659932 0 7 3.13400675 7 7 0 5-7 11-7 11zm0-8c-1.65685425 0-3-1.34314575-3-3s1.34314575-3 3-3c1.6568542 0 3 1.34314575 3 3s-1.3431458 3-3 3zm0-1c1.1045695 0 2-.8954305 2-2s-.8954305-2-2-2-2 .8954305-2 2 .8954305 2 2 2z" fill-rule="evenodd"/></symbol><symbol id="icon-minus" viewBox="0 0 16 16"><path d="m2.00087166 7h11.99825664c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-11.99825664c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-newsletter" viewBox="0 0 18 18"><path d="m9 11.8482489 2-1.1428571v-1.7053918h-4v1.7053918zm-3-1.7142857v-2.1339632h6v2.1339632l3-1.71428574v-6.41967746h-12v6.41967746zm10-5.3839632 1.5299989.95624934c.2923814.18273835.4700011.50320827.4700011.8479983v8.44575236c0 1.1045695-.8954305 2-2 2h-14c-1.1045695 0-2-.8954305-2-2v-8.44575236c0-.34479003.1776197-.66525995.47000106-.8479983l1.52999894-.95624934v-2.75c0-.55228475.44771525-1 1-1h12c.5522847 0 1 .44771525 1 1zm0 1.17924764v3.07075236l-7 4-7-4v-3.07075236l-1 .625v8.44575236c0 .5522847.44771525 1 1 1h14c.5522847 0 1-.4477153 1-1v-8.44575236zm-10-1.92924764h6v1h-6zm-1 2h8v1h-8z" fill-rule="evenodd"/></symbol><symbol id="icon-orcid" viewBox="0 0 18 18"><path d="m9 1c4.418278 0 8 3.581722 8 8s-3.581722 8-8 8-8-3.581722-8-8 3.581722-8 8-8zm-2.90107518 5.2732337h-1.41865256v7.1712107h1.41865256zm4.55867178.02508949h-2.99247027v7.14612121h2.91062487c.7673039 0 1.4476365-.1483432 2.0410182-.445034s1.0511995-.7152915 1.3734671-1.2558144c.3222677-.540523.4833991-1.1603247.4833991-1.85942385 0-.68545815-.1602789-1.30270225-.4808414-1.85175082-.3205625-.54904856-.7707074-.97532211-1.3504481-1.27883343-.5797408-.30351132-1.2413173-.45526471-1.9847495-.45526471zm-.1892674 1.07933542c.7877654 0 1.4143875.22336734 1.8798852.67010873.4654977.44674138.698243 1.05546001.698243 1.82617415 0 .74343221-.2310402 1.34447791-.6931277 1.80315511-.4620874.4586773-1.0750688.6880124-1.8389625.6880124h-1.46810075v-4.98745039zm-5.08652545-3.71099194c-.21825533 0-.410525.08444276-.57681478.25333081-.16628977.16888806-.24943341.36245684-.24943341.58071218 0 .22345188.08314364.41961891.24943341.58850696.16628978.16888806.35855945.25333082.57681478.25333082.233845 0 .43390938-.08314364.60019916-.24943342.16628978-.16628977.24943342-.36375592.24943342-.59240436 0-.233845-.08314364-.43131115-.24943342-.59240437s-.36635416-.24163862-.60019916-.24163862z" fill-rule="evenodd"/></symbol><symbol id="icon-plus" viewBox="0 0 16 16"><path d="m2.00087166 7h4.99912834v-4.99912834c0-.55276616.44386482-1.00087166 1-1.00087166.55228475 0 1 .44463086 1 1.00087166v4.99912834h4.9991283c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-4.9991283v4.9991283c0 .5527662-.44386482 1.0008717-1 1.0008717-.55228475 0-1-.4446309-1-1.0008717v-4.9991283h-4.99912834c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-print" viewBox="0 0 18 18"><path d="m16.0049107 5h-14.00982141c-.54941618 0-.99508929.4467783-.99508929.99961498v6.00077002c0 .5570958.44271433.999615.99508929.999615h1.00491071v-3h12v3h1.0049107c.5494162 0 .9950893-.4467783.9950893-.999615v-6.00077002c0-.55709576-.4427143-.99961498-.9950893-.99961498zm-2.0049107-1v-2.00208688c0-.54777062-.4519464-.99791312-1.0085302-.99791312h-7.9829396c-.55661731 0-1.0085302.44910695-1.0085302.99791312v2.00208688zm1 10v2.0018986c0 1.103521-.9019504 1.9981014-2.0085302 1.9981014h-7.9829396c-1.1092806 0-2.0085302-.8867064-2.0085302-1.9981014v-2.0018986h-1.00491071c-1.10185739 0-1.99508929-.8874333-1.99508929-1.999615v-6.00077002c0-1.10435686.8926228-1.99961498 1.99508929-1.99961498h1.00491071v-2.00208688c0-1.10341695.90195036-1.99791312 2.0085302-1.99791312h7.9829396c1.1092806 0 2.0085302.89826062 2.0085302 1.99791312v2.00208688h1.0049107c1.1018574 0 1.9950893.88743329 1.9950893 1.99961498v6.00077002c0 1.1043569-.8926228 1.999615-1.9950893 1.999615zm-1-3h-10v5.0018986c0 .5546075.44702548.9981014 1.0085302.9981014h7.9829396c.5565964 0 1.0085302-.4491701 1.0085302-.9981014zm-9 1h8v1h-8zm0 2h5v1h-5zm9-5c-.5522847 0-1-.44771525-1-1s.4477153-1 1-1 1 .44771525 1 1-.4477153 1-1 1z" fill-rule="evenodd"/></symbol><symbol id="icon-search" viewBox="0 0 22 22"><path d="M21.697 20.261a1.028 1.028 0 01.01 1.448 1.034 1.034 0 01-1.448-.01l-4.267-4.267A9.812 9.811 0 010 9.812a9.812 9.811 0 1117.43 6.182zM9.812 18.222A8.41 8.41 0 109.81 1.403a8.41 8.41 0 000 16.82z" fill-rule="evenodd"/></symbol><symbol id="icon-social-facebook" viewBox="0 0 24 24"><path d="m6.00368507 20c-1.10660471 0-2.00368507-.8945138-2.00368507-1.9940603v-12.01187942c0-1.10128908.89451376-1.99406028 1.99406028-1.99406028h12.01187942c1.1012891 0 1.9940603.89451376 1.9940603 1.99406028v12.01187942c0 1.1012891-.88679 1.9940603-2.0032184 1.9940603h-2.9570132v-6.1960818h2.0797387l.3114113-2.414723h-2.39115v-1.54164807c0-.69911803.1941355-1.1755439 1.1966615-1.1755439l1.2786739-.00055875v-2.15974763l-.2339477-.02492088c-.3441234-.03134957-.9500153-.07025255-1.6293054-.07025255-1.8435726 0-3.1057323 1.12531866-3.1057323 3.19187953v1.78079225h-2.0850778v2.414723h2.0850778v6.1960818z" fill-rule="evenodd"/></symbol><symbol id="icon-social-twitter" viewBox="0 0 24 24"><path d="m18.8767135 6.87445248c.7638174-.46908424 1.351611-1.21167363 1.6250764-2.09636345-.7135248.43394112-1.50406.74870123-2.3464594.91677702-.6695189-.73342162-1.6297913-1.19486605-2.6922204-1.19486605-2.0399895 0-3.6933555 1.69603749-3.6933555 3.78628909 0 .29642457.0314329.58673729.0942985.8617704-3.06469922-.15890802-5.78835241-1.66547825-7.60988389-3.9574208-.3174714.56076194-.49978171 1.21167363-.49978171 1.90536824 0 1.31404706.65223085 2.47224203 1.64236444 3.15218497-.60350999-.0198635-1.17401554-.1925232-1.67222562-.47366811v.04583885c0 1.83355406 1.27302891 3.36609966 2.96411421 3.71294696-.31118484.0886217-.63651445.1329326-.97441718.1329326-.2357461 0-.47149219-.0229194-.69466516-.0672303.47149219 1.5065703 1.83253297 2.6036468 3.44975116 2.632678-1.2651707 1.0160946-2.85724264 1.6196394-4.5891906 1.6196394-.29861172 0-.59093688-.0152796-.88011875-.0504227 1.63450624 1.0726291 3.57548241 1.6990934 5.66104951 1.6990934 6.79263079 0 10.50641749-5.7711113 10.50641749-10.7751859l-.0094298-.48894775c.7229547-.53478659 1.3516109-1.20250585 1.8419628-1.96190282-.6632323.30100846-1.3751855.50422736-2.1217148.59590507z" fill-rule="evenodd"/></symbol><symbol id="icon-social-youtube" viewBox="0 0 24 24"><path d="m10.1415 14.3973208-.0005625-5.19318431 4.863375 2.60554491zm9.963-7.92753362c-.6845625-.73643756-1.4518125-.73990314-1.803375-.7826454-2.518875-.18714178-6.2971875-.18714178-6.2971875-.18714178-.007875 0-3.7861875 0-6.3050625.18714178-.352125.04274226-1.1188125.04620784-1.8039375.7826454-.5394375.56084773-.7149375 1.8344515-.7149375 1.8344515s-.18 1.49597903-.18 2.99138042v1.4024082c0 1.495979.18 2.9913804.18 2.9913804s.1755 1.2736038.7149375 1.8344515c.685125.7364376 1.5845625.7133337 1.9850625.7901542 1.44.1420891 6.12.1859866 6.12.1859866s3.78225-.005776 6.301125-.1929178c.3515625-.0433198 1.1188125-.0467854 1.803375-.783223.5394375-.5608477.7155-1.8344515.7155-1.8344515s.18-1.4954014.18-2.9913804v-1.4024082c0-1.49540139-.18-2.99138042-.18-2.99138042s-.1760625-1.27360377-.7155-1.8344515z" fill-rule="evenodd"/></symbol><symbol id="icon-subject-medicine" viewBox="0 0 18 18"><path d="m12.5 8h-6.5c-1.65685425 0-3 1.34314575-3 3v1c0 1.6568542 1.34314575 3 3 3h1v-2h-.5c-.82842712 0-1.5-.6715729-1.5-1.5s.67157288-1.5 1.5-1.5h1.5 2 1 2c1.6568542 0 3-1.34314575 3-3v-1c0-1.65685425-1.3431458-3-3-3h-2v2h1.5c.8284271 0 1.5.67157288 1.5 1.5s-.6715729 1.5-1.5 1.5zm-5.5-1v-1h-3.5c-1.38071187 0-2.5-1.11928813-2.5-2.5s1.11928813-2.5 2.5-2.5h1.02786405c.46573528 0 .92507448.10843528 1.34164078.31671843l1.13382424.56691212c.06026365-1.05041141.93116291-1.88363055 1.99667093-1.88363055 1.1045695 0 2 .8954305 2 2h2c2.209139 0 4 1.790861 4 4v1c0 2.209139-1.790861 4-4 4h-2v1h2c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2h-2c0 1.1045695-.8954305 2-2 2s-2-.8954305-2-2h-1c-2.209139 0-4-1.790861-4-4v-1c0-2.209139 1.790861-4 4-4zm0-2v-2.05652691c-.14564246-.03538148-.28733393-.08714006-.42229124-.15461871l-1.15541752-.57770876c-.27771087-.13885544-.583937-.21114562-.89442719-.21114562h-1.02786405c-.82842712 0-1.5.67157288-1.5 1.5s.67157288 1.5 1.5 1.5zm4 1v1h1.5c.2761424 0 .5-.22385763.5-.5s-.2238576-.5-.5-.5zm-1 1v-5c0-.55228475-.44771525-1-1-1s-1 .44771525-1 1v5zm-2 4v5c0 .5522847.44771525 1 1 1s1-.4477153 1-1v-5zm3 2v2h2c.5522847 0 1-.4477153 1-1s-.4477153-1-1-1zm-4-1v-1h-.5c-.27614237 0-.5.2238576-.5.5s.22385763.5.5.5zm-3.5-9h1c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-success" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm3.4860198 4.98163161-4.71802968 5.50657859-2.62834168-2.02300024c-.42862421-.36730544-1.06564993-.30775346-1.42283677.13301307-.35718685.44076653-.29927542 1.0958383.12934879 1.46314377l3.40735508 2.7323063c.42215801.3385221 1.03700951.2798252 1.38749189-.1324571l5.38450527-6.33394549c.3613513-.43716226.3096573-1.09278382-.115462-1.46437175-.4251192-.37158792-1.0626796-.31842941-1.4240309.11873285z" fill-rule="evenodd"/></symbol><symbol id="icon-table" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587l-4.0059107-.001.001.001h-1l-.001-.001h-5l.001.001h-1l-.001-.001-3.00391071.001c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm-11.0059107 5h-3.999v6.9941413c0 .5572961.44630695 1.0058587.99508929 1.0058587h3.00391071zm6 0h-5v8h5zm5.0059107-4h-4.0059107v3h5.001v1h-5.001v7.999l4.0059107.001c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-12.5049107 9c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.2238576.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238576-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm-6-2c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.2238576.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238576-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm-6-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.22385763-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm1.499-5h-5v3h5zm-6 0h-3.00391071c-.54871518 0-.99508929.44887827-.99508929 1.00585866v1.99414134h3.999z" fill-rule="evenodd"/></symbol><symbol id="icon-tick-circle" viewBox="0 0 24 24"><path d="m12 2c5.5228475 0 10 4.4771525 10 10s-4.4771525 10-10 10-10-4.4771525-10-10 4.4771525-10 10-10zm0 1c-4.97056275 0-9 4.02943725-9 9 0 4.9705627 4.02943725 9 9 9 4.9705627 0 9-4.0294373 9-9 0-4.97056275-4.0294373-9-9-9zm4.2199868 5.36606669c.3613514-.43716226.9989118-.49032077 1.424031-.11873285s.4768133 1.02720949.115462 1.46437175l-6.093335 6.94397871c-.3622945.4128716-.9897871.4562317-1.4054264.0971157l-3.89719065-3.3672071c-.42862421-.3673054-.48653564-1.0223772-.1293488-1.4631437s.99421256-.5003185 1.42283677-.1330131l3.11097438 2.6987741z" fill-rule="evenodd"/></symbol><symbol id="icon-tick" viewBox="0 0 16 16"><path d="m6.76799012 9.21106946-3.1109744-2.58349728c-.42862421-.35161617-1.06564993-.29460792-1.42283677.12733148s-.29927541 1.04903009.1293488 1.40064626l3.91576307 3.23873978c.41034319.3393961 1.01467563.2976897 1.37450571-.0948578l6.10568327-6.660841c.3613513-.41848908.3096572-1.04610608-.115462-1.4018218-.4251192-.35571573-1.0626796-.30482786-1.424031.11366122z" fill-rule="evenodd"/></symbol><symbol id="icon-update" viewBox="0 0 18 18"><path d="m1 13v1c0 .5522847.44771525 1 1 1h14c.5522847 0 1-.4477153 1-1v-1h-1v-10h-14v10zm16-1h1v2c0 1.1045695-.8954305 2-2 2h-14c-1.1045695 0-2-.8954305-2-2v-2h1v-9c0-.55228475.44771525-1 1-1h14c.5522847 0 1 .44771525 1 1zm-1 0v1h-4.5857864l-1 1h-2.82842716l-1-1h-4.58578644v-1h5l1 1h2l1-1zm-13-8h12v7h-12zm1 1v5h10v-5zm1 1h4v1h-4zm0 2h4v1h-4z" fill-rule="evenodd"/></symbol><symbol id="icon-upload" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm0 1h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v14.00982141c0 .5500396.44491393.9950893.99406028.9950893h12.01187942c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717zm-1.85576936 4.14572769c.19483374-.19483375.51177826-.19377714.70556874.00001334l2.59099082 2.59099079c.1948411.19484112.1904373.51514474.0027906.70279143-.1932998.19329987-.5046517.19237083-.7001856-.00692852l-1.74638687-1.7800176v6.14827687c0 .2717771-.23193359.492096-.5.492096-.27614237 0-.5-.216372-.5-.492096v-6.14827641l-1.74627892 1.77990922c-.1933927.1971171-.51252214.19455839-.70016883.0069117-.19329987-.19329988-.19100584-.50899493.00277731-.70277808z" fill-rule="evenodd"/></symbol><symbol id="icon-video" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587h-14.00982141c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm0 1h-14.00982141c-.54871518 0-.99508929.44887827-.99508929 1.00585866v9.98828264c0 .5572961.44630695 1.0058587.99508929 1.0058587h14.00982141c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-8.30912922 2.24944486 4.60460462 2.73982242c.9365543.55726659.9290753 1.46522435 0 2.01804082l-4.60460462 2.7398224c-.93655425.5572666-1.69578148.1645632-1.69578148-.8937585v-5.71016863c0-1.05087579.76670616-1.446575 1.69578148-.89375851zm-.67492769.96085624v5.5750128c0 .2995102-.10753745.2442517.16578928.0847713l4.58452283-2.67497259c.3050619-.17799716.3051624-.21655446 0-.39461026l-4.58452283-2.67497264c-.26630747-.15538481-.16578928-.20699944-.16578928.08477139z" fill-rule="evenodd"/></symbol><symbol id="icon-warning" viewBox="0 0 18 18"><path d="m9 11.75c.69035594 0 1.25.5596441 1.25 1.25s-.55964406 1.25-1.25 1.25-1.25-.5596441-1.25-1.25.55964406-1.25 1.25-1.25zm.41320045-7.75c.55228475 0 1.00000005.44771525 1.00000005 1l-.0034543.08304548-.3333333 4c-.043191.51829212-.47645714.91695452-.99654578.91695452h-.15973424c-.52008864 0-.95335475-.3986624-.99654576-.91695452l-.33333333-4c-.04586475-.55037702.36312325-1.03372649.91350028-1.07959124l.04148683-.00259031zm-.41320045 14c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-left-bullet" viewBox="0 0 8 16"><path d="M3 8l5 5v3L0 8l8-8v3L3 8z"/></symbol><symbol id="icon-chevron-down" viewBox="0 0 16 16"><path d="m5.58578644 3-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4c-.39052429.39052429-1.02368927.39052429-1.41421356 0s-.39052429-1.02368927 0-1.41421356z" fill-rule="evenodd" transform="matrix(0 1 -1 0 11 3)"/></symbol><symbol id="icon-download-rounded"><path d="M0 13c0-.556.449-1 1.002-1h9.996a.999.999 0 110 2H1.002A1.006 1.006 0 010 13zM7 1v6.8l2.482-2.482c.392-.392 1.022-.4 1.403-.02a1.001 1.001 0 010 1.417l-4.177 4.177a1.001 1.001 0 01-1.416 0L1.115 6.715a.991.991 0 01-.016-1.4 1 1 0 011.42.003L5 7.8V1c0-.55.444-.996 1-.996.552 0 1 .445 1 .996z"/></symbol><symbol id="icon-ext-link" viewBox="0 0 16 16"><path d="M12.9 16H3.1C1.4 16 0 14.6 0 12.9V3.2C0 1.4 1.4 0 3.1 0h3.7v1H3.1C2 1 1 2 1 3.2v9.7C1 14 2 15 3.1 15h9.7c1.2 0 2.1-1 2.1-2.1V8.7h1v4.2c.1 1.7-1.3 3.1-3 3.1z"/><path d="M12.8 2.5l.7.7-9 8.9-.7-.7 9-8.9z"/><path d="M9.7 0L16 6.2V0z"/></symbol><symbol id="icon-remove" viewBox="-296 388 18 18"><path d="M-291.7 396.1h9v2h-9z"/><path d="M-287 405.5c-4.7 0-8.5-3.8-8.5-8.5s3.8-8.5 8.5-8.5 8.5 3.8 8.5 8.5-3.8 8.5-8.5 8.5zm0-16c-4.1 0-7.5 3.4-7.5 7.5s3.4 7.5 7.5 7.5 7.5-3.4 7.5-7.5-3.4-7.5-7.5-7.5z"/></symbol><symbol id="icon-rss" viewBox="0 0 18 18"><path d="m.97480857 6.01583891.11675372.00378391c5.75903295.51984988 10.34261021 5.10537458 10.85988231 10.86480098.0494035.5500707-.3564674 1.0360406-.906538 1.0854441-.5500707.0494036-1.0360406-.3564673-1.08544412-.906538-.43079083-4.7965248-4.25151132-8.61886853-9.04770289-9.05180573-.55004837-.04965115-.95570047-.53580366-.90604933-1.08585203.04610464-.5107592.46858035-.89701345.96909831-.90983323zm1.52519143 6.95474179c1.38071187 0 2.5 1.1192881 2.5 2.5s-1.11928813 2.5-2.5 2.5-2.5-1.1192881-2.5-2.5 1.11928813-2.5 2.5-2.5zm-1.43253846-12.96884168c9.09581416.53242539 16.37540296 7.8163886 16.90205336 16.91294558.0319214.5513615-.389168 1.0242056-.9405294 1.056127-.5513615.0319214-1.0242057-.389168-1.0561271-.9405294-.4679958-8.08344784-6.93949306-14.55883389-15.02226722-15.03196077-.55134101-.03227286-.97212889-.50538538-.93985602-1.05672639.03227286-.551341.50538538-.97212888 1.05672638-.93985602z" fill-rule="evenodd"/></symbol><symbol id="icon-springer-arrow-left"><path d="M15 7a1 1 0 000-2H3.385l2.482-2.482a.994.994 0 00.02-1.403 1.001 1.001 0 00-1.417 0L.294 5.292a1.001 1.001 0 000 1.416l4.176 4.177a.991.991 0 001.4.016 1 1 0 00-.003-1.42L3.385 7H15z"/></symbol><symbol id="icon-springer-arrow-right"><path d="M1 7a1 1 0 010-2h11.615l-2.482-2.482a.994.994 0 01-.02-1.403 1.001 1.001 0 011.417 0l4.176 4.177a1.001 1.001 0 010 1.416l-4.176 4.177a.991.991 0 01-1.4.016 1 1 0 01.003-1.42L12.615 7H1z"/></symbol><symbol id="icon-springer-collections" viewBox="3 3 32 32"><path fill-rule="evenodd" d="M25.583333,30.1249997 L25.583333,7.1207574 C25.583333,7.10772495 25.579812,7.10416665 25.5859851,7.10416665 L6.10400517,7.10571021 L6.10400517,30.1355179 C6.10400517,31.1064087 6.89406744,31.8958329 7.86448169,31.8958329 L26.057145,31.8958329 C25.7558021,31.374901 25.583333,30.7700915 25.583333,30.1249997 Z M4.33333333,30.1355179 L4.33333333,7.10571021 C4.33333333,6.12070047 5.12497502,5.33333333 6.10151452,5.33333333 L25.5859851,5.33333333 C26.5617372,5.33333333 27.3541664,6.13359035 27.3541664,7.1207574 L27.3541664,12.4166666 L32.6666663,12.4166666 L32.6666663,30.1098941 C32.6666663,32.0694626 31.0857174,33.6666663 29.1355179,33.6666663 L7.86448169,33.6666663 C5.91736809,33.6666663 4.33333333,32.0857174 4.33333333,30.1355179 Z M27.3541664,14.1874999 L27.3541664,30.1249997 C27.3541664,31.1030039 28.1469954,31.8958329 29.1249997,31.8958329 C30.1030039,31.8958329 30.8958329,31.1030039 30.8958329,30.1249997 L30.8958329,14.1874999 L27.3541664,14.1874999 Z M9.64583326,10.6458333 L22.0416665,10.6458333 L22.0416665,17.7291665 L9.64583326,17.7291665 L9.64583326,10.6458333 Z M11.4166666,12.4166666 L11.4166666,15.9583331 L20.2708331,15.9583331 L20.2708331,12.4166666 L11.4166666,12.4166666 Z M9.64583326,19.4999998 L22.0416665,19.4999998 L22.0416665,21.2708331 L9.64583326,21.2708331 L9.64583326,19.4999998 Z M9.64583326,23.0416665 L22.0416665,23.0416665 L22.0416665,24.8124997 L9.64583326,24.8124997 L9.64583326,23.0416665 Z M9.64583326,26.583333 L22.0416665,26.583333 L22.0416665,28.3541664 L9.64583326,28.3541664 L9.64583326,26.583333 Z"/></symbol><symbol id="icon-springer-download" viewBox="-301 390 9 14"><path d="M-301 395.6l4.5 5.1 4.5-5.1h-3V390h-3v5.6h-3zm0 6.5h9v1.9h-9z"/></symbol><symbol id="icon-springer-info" viewBox="0 0 24 24"><g id="V&I" stroke="none" stroke-width="1" fill-rule="evenodd"><g id="info" fill-rule="nonzero"><path d="M12,0 C18.627417,0 24,5.372583 24,12 C24,18.627417 18.627417,24 12,24 C5.372583,24 0,18.627417 0,12 C0,5.372583 5.372583,0 12,0 Z M12.5540543,9.1 L11.5540543,9.1 C11.0017696,9.1 10.5540543,9.54771525 10.5540543,10.1 L10.5540543,10.1 L10.5540543,18.1 C10.5540543,18.6522847 11.0017696,19.1 11.5540543,19.1 L11.5540543,19.1 L12.5540543,19.1 C13.1063391,19.1 13.5540543,18.6522847 13.5540543,18.1 L13.5540543,18.1 L13.5540543,10.1 C13.5540543,9.54771525 13.1063391,9.1 12.5540543,9.1 L12.5540543,9.1 Z M12,5 C11.5356863,5 11.1529412,5.14640523 10.8517647,5.43921569 C10.5505882,5.73202614 10.4,6.11546841 10.4,6.58954248 C10.4,7.06361656 10.5505882,7.45054466 10.8517647,7.7503268 C11.1529412,8.05010893 11.5356863,8.2 12,8.2 C12.4768627,8.2 12.8627451,8.05010893 13.1576471,7.7503268 C13.452549,7.45054466 13.6,7.06361656 13.6,6.58954248 C13.6,6.11546841 13.452549,5.73202614 13.1576471,5.43921569 C12.8627451,5.14640523 12.4768627,5 12,5 Z" id="Combined-Shape"/></g></g></symbol><symbol id="icon-springer-tick-circle" viewBox="0 0 24 24"><g id="Page-1" stroke="none" stroke-width="1" fill-rule="evenodd"><g id="springer-tick-circle" fill-rule="nonzero"><path d="M12,24 C5.372583,24 0,18.627417 0,12 C0,5.372583 5.372583,0 12,0 C18.627417,0 24,5.372583 24,12 C24,18.627417 18.627417,24 12,24 Z M7.657,10.79 C7.45285634,10.6137568 7.18569967,10.5283283 6.91717333,10.5534259 C6.648647,10.5785236 6.40194824,10.7119794 6.234,10.923 C5.87705269,11.3666969 5.93445559,12.0131419 6.364,12.387 L10.261,15.754 C10.6765468,16.112859 11.3037113,16.0695601 11.666,15.657 L17.759,8.713 C18.120307,8.27302248 18.0695334,7.62621189 17.644,7.248 C17.4414817,7.06995024 17.1751516,6.9821166 16.9064461,7.00476032 C16.6377406,7.02740404 16.3898655,7.15856958 16.22,7.368 L10.768,13.489 L7.657,10.79 Z" id="path-1"/></g></g></symbol><symbol id="icon-updates" viewBox="0 0 18 18"><g fill-rule="nonzero"><path d="M16.98 3.484h-.48c-2.52-.058-5.04 1.161-7.44 2.903-2.46-1.8-4.74-2.903-8.04-2.903-.3 0-.54.29-.54.58v9.813c0 .29.24.523.54.581 2.76.348 4.86 1.045 7.62 2.903.24.116.54.116.72 0 2.76-1.858 4.86-2.555 7.62-2.903.3-.058.54-.29.54-.58V4.064c0-.29-.24-.523-.54-.581zm-15.3 1.22c2.34 0 4.86 1.509 6.72 2.786v8.478c-2.34-1.394-4.38-2.09-6.72-2.439V4.703zm14.58 8.767c-2.34.348-4.38 1.045-6.72 2.439V7.374C12 5.632 14.1 4.645 16.26 4.645v8.826z"/><path d="M9 .058c-1.56 0-2.76 1.22-2.76 2.671C6.24 4.181 7.5 5.4 9 5.4c1.5 0 2.76-1.22 2.76-2.671 0-1.452-1.2-2.67-2.76-2.67zm0 4.413c-.96 0-1.8-.755-1.8-1.742C7.2 1.8 7.98.987 9 .987s1.8.755 1.8 1.742c0 .93-.84 1.742-1.8 1.742z"/></g></symbol><symbol id="icon-checklist-banner" viewBox="0 0 56.69 56.69"><path style="fill:none" d="M0 0h56.69v56.69H0z"/><clipPath id="b"><use xlink:href="#a" style="overflow:visible"/></clipPath><path d="M21.14 34.46c0-6.77 5.48-12.26 12.24-12.26s12.24 5.49 12.24 12.26-5.48 12.26-12.24 12.26c-6.76-.01-12.24-5.49-12.24-12.26zm19.33 10.66 10.23 9.22s1.21 1.09 2.3-.12l2.09-2.32s1.09-1.21-.12-2.3l-10.23-9.22m-19.29-5.92c0-4.38 3.55-7.94 7.93-7.94s7.93 3.55 7.93 7.94c0 4.38-3.55 7.94-7.93 7.94-4.38-.01-7.93-3.56-7.93-7.94zm17.58 12.99 4.14-4.81" style="clip-path:url(#b);fill:none;stroke:#01324b;stroke-width:2;stroke-linecap:round"/><path d="M8.26 9.75H28.6M8.26 15.98H28.6m-20.34 6.2h12.5m14.42-5.2V4.86s0-2.93-2.93-2.93H4.13s-2.93 0-2.93 2.93v37.57s0 2.93 2.93 2.93h15.01M8.26 9.75H28.6M8.26 15.98H28.6m-20.34 6.2h12.5" style="clip-path:url(#b);fill:none;stroke:#01324b;stroke-width:2;stroke-linecap:round;stroke-linejoin:round"/></symbol><symbol id="icon-submit-closed" viewBox="0 0 18 18"><path d="m15 0c1.1045695 0 2 .8954305 2 2v4.5c0 .27614237-.2238576.5-.5.5s-.5-.22385763-.5-.5v-4.5c0-.51283584-.3860402-.93550716-.8833789-.99327227l-.1166211-.00672773h-9v3c0 1.1045695-.8954305 2-2 2h-3v10c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h4.5c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-4.5c-1.1045695 0-2-.8954305-2-2v-10.17157288c0-.53043297.21071368-1.0391408.58578644-1.41421356l3.82842712-3.82842712c.37507276-.37507276.88378059-.58578644 1.41421356-.58578644zm-2.5 7c3.0375661 0 5.5 2.46243388 5.5 5.5 0 3.0375661-2.4624339 5.5-5.5 5.5-3.03756612 0-5.5-2.4624339-5.5-5.5 0-3.03756612 2.46243388-5.5 5.5-5.5zm0 1c-2.4852814 0-4.5 2.0147186-4.5 4.5s2.0147186 4.5 4.5 4.5 4.5-2.0147186 4.5-4.5-2.0147186-4.5-4.5-4.5zm2.3087379 2.1912621c.2550161.2550162.2550161.668479 0 .9234952l-1.3859024 1.3845831 1.3859024 1.3859023c.2550161.2550162.2550161.668479 0 .9234952-.2550162.2550161-.668479.2550161-.9234952 0l-1.3859023-1.3859024-1.3845831 1.3859024c-.2550162.2550161-.668479.2550161-.9234952 0-.25501614-.2550162-.25501614-.668479 0-.9234952l1.3845831-1.3859023-1.3845831-1.3845831c-.25501614-.2550162-.25501614-.668479 0-.9234952.2550162-.25501614.668479-.25501614.9234952 0l1.3845831 1.3845831 1.3859023-1.3845831c.2550162-.25501614.668479-.25501614.9234952 0zm-9.8087379-8.7782621-3.587 3.587h2.587c.55228475 0 1-.44771525 1-1z"/></symbol><symbol id="icon-submit-open" viewBox="0 0 18 18"><path d="m15 0c1.1045695 0 2 .8954305 2 2v5.5c0 .27614237-.2238576.5-.5.5s-.5-.22385763-.5-.5v-5.5c0-.51283584-.3860402-.93550716-.8833789-.99327227l-.1166211-.00672773h-9v3c0 1.1045695-.8954305 2-2 2h-3v10c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h7.5c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-7.5c-1.1045695 0-2-.8954305-2-2v-10.17157288c0-.53043297.21071368-1.0391408.58578644-1.41421356l3.82842712-3.82842712c.37507276-.37507276.88378059-.58578644 1.41421356-.58578644zm-.5442863 8.18867991 3.3545404 3.35454039c.2508994.2508994.2538696.6596433.0035959.909917-.2429543.2429542-.6561449.2462671-.9065387-.0089489l-2.2609825-2.3045251.0010427 7.2231989c0 .3569916-.2898381.6371378-.6473715.6371378-.3470771 0-.6473715-.2852563-.6473715-.6371378l-.0010428-7.2231995-2.2611222 2.3046654c-.2531661.2580415-.6562868.2592444-.9065605.0089707-.24295423-.2429542-.24865597-.6576651.0036132-.9099343l3.3546673-3.35466731c.2509089-.25090888.6612706-.25227691.9135302-.00001728zm-.9557137-3.18867991c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-6c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm-8.5-3.587-3.587 3.587h2.587c.55228475 0 1-.44771525 1-1zm8.5 1.587c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-6c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z"/></symbol><symbol id="icon-submit-upcoming" viewBox="0 0 18 18"><path d="m15 0c1.1045695 0 2 .8954305 2 2v4.5c0 .27614237-.2238576.5-.5.5s-.5-.22385763-.5-.5v-4.5c0-.51283584-.3860402-.93550716-.8833789-.99327227l-.1166211-.00672773h-9v3c0 1.1045695-.8954305 2-2 2h-3v10c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h4.5c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-4.5c-1.1045695 0-2-.8954305-2-2v-10.17157288c0-.53043297.21071368-1.0391408.58578644-1.41421356l3.82842712-3.82842712c.37507276-.37507276.88378059-.58578644 1.41421356-.58578644zm-2.5 7c3.0375661 0 5.5 2.46243388 5.5 5.5 0 3.0375661-2.4624339 5.5-5.5 5.5-3.03756612 0-5.5-2.4624339-5.5-5.5 0-1.6607442.73606908-3.14957021 1.89976608-4.15803695l-1.51549374.02214397c-.27613212.00263356-.49998143-.22483432-.49998143-.49020681 0-.24299316.17766103-.44509007.40961587-.48700057l.08928713-.00797472h2.66407569c.2449213 0 .4486219.17766776.490865.40963137l.008038.08929051v2.6642143c0 .275547-.2296028.4989219-.4949753.4989219-.24299317 0-.44342617-.1744719-.4830969-.4093269l-.00710993-.0906783.01983146-1.46576707c-.96740882.82538117-1.58082193 2.05345007-1.58082193 3.42478927 0 2.4852814 2.0147186 4.5 4.5 4.5s4.5-2.0147186 4.5-4.5-2.0147186-4.5-4.5-4.5c-.7684937 0-.7684937-1 0-1zm0 2.85c.3263501 0 .5965265.2405082.6429523.5539478l.0070477.0960522v1.731l.8096194.8093806c.2284567.2284567.2513024.5846637.068537.8386705l-.068537.0805683c-.2284567.2284567-.5846637.2513024-.8386705.068537l-.0805683-.068537-.9707107-.9707107c-.1125218-.1125218-.1855975-.257116-.2103268-.412296l-.0093431-.1180341v-1.9585786c0-.3589851.2910149-.65.65-.65zm-7.5-8.437-3.587 3.587h2.587c.55228475 0 1-.44771525 1-1z"/></symbol><symbol id="icon-facebook-bordered" viewBox="463.812 263.868 32 32"><path d="M479.812,263.868c-8.837,0-16,7.163-16,16s7.163,16,16,16s16-7.163,16-16S488.649,263.868,479.812,263.868z M479.812,293.868c-7.732,0-14-6.269-14-14s6.268-14,14-14s14,6.269,14,14S487.545,293.868,479.812,293.868z"/><path d="M483.025,280.48l0.32-2.477h-2.453v-1.582c0-0.715,0.199-1.207,1.227-1.207h1.311v-2.213 c-0.227-0.029-1.003-0.098-1.907-0.098c-1.894,0-3.186,1.154-3.186,3.271v1.826h-2.142v2.477h2.142v6.354h2.557v-6.354 L483.025,280.48L483.025,280.48z"/></symbol><symbol id="icon-twitter-bordered" viewBox="463.812 263.868 32 32"><g><path d="M486.416,276.191c-0.483,0.215-1.007,0.357-1.554,0.429c0.558-0.338,0.991-0.868,1.19-1.502 c-0.521,0.308-1.104,0.536-1.72,0.657c-0.494-0.526-1.2-0.854-1.979-0.854c-1.496,0-2.711,1.213-2.711,2.71 c0,0.212,0.023,0.419,0.069,0.616c-2.252-0.111-4.25-1.19-5.586-2.831c-0.231,0.398-0.365,0.866-0.365,1.361 c0,0.94,0.479,1.772,1.204,2.257c-0.441-0.015-0.861-0.138-1.227-0.339v0.031c0,1.314,0.937,2.41,2.174,2.656 c-0.227,0.062-0.47,0.098-0.718,0.098c-0.171,0-0.343-0.018-0.511-0.049c0.35,1.074,1.347,1.859,2.531,1.883 c-0.928,0.726-2.095,1.16-3.366,1.16c-0.22,0-0.433-0.014-0.644-0.037c1.2,0.768,2.621,1.215,4.155,1.215 c4.983,0,7.71-4.129,7.71-7.711c0-0.115-0.004-0.232-0.006-0.351C485.592,277.212,486.054,276.734,486.416,276.191z"/></g><path d="M479.812,263.868c-8.837,0-16,7.163-16,16s7.163,16,16,16s16-7.163,16-16S488.649,263.868,479.812,263.868z M479.812,293.868c-7.732,0-14-6.269-14-14s6.268-14,14-14s14,6.269,14,14S487.545,293.868,479.812,293.868z"/></symbol><symbol id="icon-weibo-bordered" viewBox="463.812 263.868 32 32"><path d="M479.812,263.868c-8.838,0-16,7.163-16,16s7.162,16,16,16c8.837,0,16-7.163,16-16S488.649,263.868,479.812,263.868z M479.812,293.868c-7.732,0-14-6.269-14-14s6.268-14,14-14c7.731,0,14,6.269,14,14S487.545,293.868,479.812,293.868z"/><g><path d="M478.552,285.348c-2.616,0.261-4.876-0.926-5.044-2.649c-0.167-1.722,1.814-3.33,4.433-3.588 c2.609-0.263,4.871,0.926,5.041,2.647C483.147,283.479,481.164,285.089,478.552,285.348 M483.782,279.63 c-0.226-0.065-0.374-0.109-0.259-0.403c0.25-0.639,0.276-1.188,0.005-1.581c-0.515-0.734-1.915-0.693-3.521-0.021 c0,0-0.508,0.224-0.378-0.181c0.247-0.798,0.209-1.468-0.178-1.852c-0.87-0.878-3.194,0.032-5.183,2.027 c-1.489,1.494-2.357,3.082-2.357,4.453c0,2.619,3.354,4.213,6.631,4.213c4.297,0,7.154-2.504,7.154-4.493 C485.697,280.594,484.689,279.911,483.782,279.63"/><path d="M486.637,274.833c-1.039-1.154-2.57-1.592-3.982-1.291l0,0c-0.325,0.068-0.532,0.391-0.465,0.72 c0.068,0.328,0.391,0.537,0.72,0.466c1.005-0.215,2.092,0.104,2.827,0.92c0.736,0.818,0.938,1.939,0.625,2.918l0,0 c-0.102,0.318,0.068,0.661,0.39,0.762c0.32,0.104,0.658-0.069,0.763-0.391v-0.001C487.953,277.558,487.674,275.985,486.637,274.833 "/><path d="M485.041,276.276c-0.504-0.562-1.25-0.774-1.938-0.63c-0.279,0.06-0.461,0.339-0.396,0.621 c0.062,0.281,0.335,0.461,0.617,0.398l0,0c0.336-0.071,0.702,0.03,0.947,0.307s0.312,0.649,0.207,0.979l0,0 c-0.089,0.271,0.062,0.565,0.336,0.654c0.274,0.09,0.564-0.062,0.657-0.336C485.686,277.604,485.549,276.837,485.041,276.276"/><path d="M478.694,282.227c-0.09,0.156-0.293,0.233-0.451,0.166c-0.151-0.062-0.204-0.235-0.115-0.389 c0.093-0.155,0.284-0.229,0.44-0.168C478.725,281.892,478.782,282.071,478.694,282.227 M477.862,283.301 c-0.253,0.405-0.795,0.58-1.202,0.396c-0.403-0.186-0.521-0.655-0.27-1.051c0.248-0.39,0.771-0.566,1.176-0.393 C477.979,282.423,478.109,282.889,477.862,283.301 M478.812,280.437c-1.244-0.326-2.65,0.294-3.19,1.396 c-0.553,1.119-0.021,2.369,1.236,2.775c1.303,0.42,2.84-0.225,3.374-1.436C480.758,281.989,480.1,280.77,478.812,280.437"/></g></symbol></svg> </div> <div class="u-vh-full"> <a class="c-skip-link" href="#main-content">Skip to main content</a> <div class="u-hide u-show-following-ad"></div> <aside class="adsbox c-ad c-ad--728x90" data-component-mpu> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-LB1" data-ad-type="LB1" data-test="LB1-ad" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springer_open/advancesincontinuousanddiscretemodels/articles" data-gpt-sizes="728x90,970x90" data-gpt-targeting="pos=LB1;doi=10.1186/s13662-016-0916-1;type=article;kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition;pmc=M12031,M00009,M12007,M12066,M12147,M12155;" > <noscript> <a href="//pubads.g.doubleclick.net/gampad/jump?iu=/270604982/springer_open/advancesincontinuousanddiscretemodels/articles&sz=728x90,970x90&pos=LB1&doi=10.1186/s13662-016-0916-1&type=article&kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition&pmc=M12031,M00009,M12007,M12066,M12147,M12155&"> <img data-test="gpt-advert-fallback-img" src="//pubads.g.doubleclick.net/gampad/ad?iu=/270604982/springer_open/advancesincontinuousanddiscretemodels/articles&sz=728x90,970x90&pos=LB1&doi=10.1186/s13662-016-0916-1&type=article&kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition&pmc=M12031,M00009,M12007,M12066,M12147,M12155&" alt="Advertisement" width="728" height="90"> </a> </noscript> </div> </div> </aside> <div id="membership-message-loader-desktop" class="placeholder" data-placeholder="/placeholder/v1/membership/message"></div> <div id="top" class="u-position-relative"> <header class="c-header" data-test="publisher-header"> <div class="c-header__container"> <div class="c-header__brand u-mr-48" itemscope itemtype="http://schema.org/Organization" data-test="navbar-logo-header"> <a href="https://www.springeropen.com" itemprop="url"> <img alt="SpringerOpen" itemprop="logo" width="160" height="30" role="img" src=/static/images/springeropen/logo-springer-open-d04c3ea16c.svg> </a> </div> <div class="c-header__navigation"> <button type="button" class="c-header__link u-button-reset u-mr-24" data-expander data-expander-target="#publisher-header-search" data-expander-autofocus="firstTabbable" data-test="header-search-button" aria-controls="publisher-header-search" aria-expanded="false"> <span class="u-display-flex u-align-items-center"> <span>Search</span> <svg class="u-icon u-flex-static u-ml-8" aria-hidden="true" focusable="false"> <use xlink:href="#icon-search"></use> </svg> </span> </button> <nav> <ul class="c-header__menu" data-enhanced-menu data-test="publisher-navigation"> <li class="c-header__item u-hide-at-lt-lg"> <a class="c-header__link" href="//www.springeropen.com/get-published"> Get published </a> </li> <li class="c-header__item u-hide-at-lt-lg"> <a class="c-header__link" href="//www.springeropen.com/journals"> Explore Journals </a> </li> <li class="c-header__item u-hide-at-lt-lg"> <a class="c-header__link" href="https://www.springer.com/gp/open-access/books"> Books </a> </li> <li class="c-header__item u-hide-at-lt-lg"> <a class="c-header__link" href="//www.springeropen.com/about"> About </a> </li> <li class="c-header__item"> <a data-enhanced-account class="c-header__link" href="https://www.springeropen.com/account" data-test="login-link"> My account </a> </li> </ul> </nav> </div> </div> </header> <div class="c-popup-search u-js-hide" id="publisher-header-search"> <div class="u-container"> <div class="c-popup-search__container"> <div class="ctx-search"> <form role="search" class="c-form-field js-skip-validation" method="GET" action="//www.springeropen.com/search" data-track="search" data-track-context="pop out website-wide search in bmc website header" data-track-category="Search and Results" data-track-action="Submit search" data-dynamic-track-label data-track-label="" data-test="global-search"> <label for="publisherSearch" class="c-form-field__label">Search all SpringerOpen articles</label> <div class="u-display-flex"> <input id="publisherSearch" class="c-form-field__input" data-search-input autocomplete="off" role="textbox" data-test="search-input" name="query" type="text" value=""/> <div> <button class="u-button u-button--primary" type="submit" data-test="search-submit-button"> <span class="u-visually-hidden">Search</span> <svg class="u-icon u-flex-static" width="16" height="16" aria-hidden="true" focusable="false"> <use xlink:href="#icon-search"></use> </svg> </button> </div> </div> <input type="hidden" name="searchType" value="publisherSearch"/> </form> </div> </div> </div> </div> </div> <header class="c-journal-header ctx-journal-header"> <div class="u-container"> <div class="c-journal-header__grid"> <div class="c-journal-header__grid-main"> <div class="h2 c-journal-header__title" id="journalTitle"> <a href="/">Advances in Continuous and Discrete Models</a> </div> <p class="c-journal-header__subtitle">Theory and Modern Applications</p> </div> </div> </div> <div class="c-navbar c-navbar--with-submit-button"> <div class="c-navbar__container"> <div class="c-navbar__content"> <nav class="c-navbar__nav"> <ul class="c-navbar__nav c-navbar__nav--journal" role="menu" data-test="site-navigation"> <li class="c-navbar__item" role="menuitem"> <a class="c-navbar__link" data-track="click" data-track-category="About" data-track-action="Clicked journal navigation link" href='/about'>About</a> </li> <li class="c-navbar__item" role="menuitem"> <a class="c-navbar__link c-navbar__link--is-shown" data-track="click" data-track-category="Articles" data-track-action="Clicked journal navigation link" href='/articles'>Articles</a> </li> <li class="c-navbar__item" role="menuitem"> <a class="c-navbar__link" data-track="click" data-track-category="Submission Guidelines" data-track-action="Clicked journal navigation link" href='/submission-guidelines'>Submission Guidelines</a> </li> <li class="c-navbar__item" role="menuitem" data-test="journal-header-submit-button"> <div class=""> <a class="u-button u-button--tertiary u-button--alt-colour-on-mobile" href="https://www.editorialmanager.com/aide/" data-track="click_submit_manuscript" data-track-action="manuscript submission" data-track-category="article" data-track-label="button in journal nav" data-track-context="journal header on article page" data-track-external data-gtm-criteo="submit-manuscript" data-test="submit-manuscript-button">Submit manuscript<svg class="u-ml-8" width="15" height="16" aria-hidden="true" focusable="false"><use xlink:href="#icon-submit-open"></use></svg></a> </div> </li> </ul> </nav> </div> </div> </div> </header> <div class="u-container u-mt-32 u-mb-32 u-clearfix" id="main-content" data-component="article-container"> <main class="c-article-main-column u-float-left js-main-column" data-track-component="article body"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container" data-track-context="sticky banner"> <div class="c-context-bar__title"> On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion </div> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both"> <a href="//advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-016-0916-1.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="link" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-download"/></svg> </a> </div> </div> </div> </div> <div class="c-pdf-button__container u-hide-at-lg js-context-bar-sticky-point-mobile"> <div class="c-pdf-container" data-track-context="article body"> <div class="c-pdf-download u-clear-both"> <a href="//advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-016-0916-1.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="link" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-download"/></svg> </a> </div> </div> </div> <article lang="en"> <div class="c-article-header"> <ul class="c-article-identifiers" data-test="article-identifier"> <li class="c-article-identifiers__item" data-test="article-category">Research</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item">Published: <time datetime="2016-07-23">23 July 2016</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Bin-Pei-Aff1" data-author-popup="auth-Bin-Pei-Aff1" data-author-search="Pei, Bin">Bin Pei</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Yong-Xu-Aff1" data-author-popup="auth-Yong-Xu-Aff1" data-author-search="Xu, Yong" data-corresp-id="c1">Yong Xu<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Advances in Difference Equations</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 2016</b>, Article number: <span data-test="article-number">194</span> (<span data-test="article-publication-year">2016</span>) <a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">2589 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/10.1186/s13662-016-0916-1/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </div> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter <span class="mathjax-tex">$H\in(\frac{1}{2},1)$</span>. The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study.</p></div></div></section> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Stochastic differential equations (SDEs) have been greatly developed and are well known to model diverse phenomena, including but not limited to fluctuating stock prices, physical systems subject to thermal fluctuations, forecasting the growth of a population, from various points of view [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title=" Øksendal, B: Stochastic Differential Equations. Springer, Berlin (2005) " href="/articles/10.1186/s13662-016-0916-1#ref-CR1" id="ref-link-section-d76369347e342">1</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title=" Gard, T: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988) " href="/articles/10.1186/s13662-016-0916-1#ref-CR4" id="ref-link-section-d76369347e345">4</a>]. There is no doubt that the mathematical models under a random disturbance of ‘Gaussian white noise’ have seen rapid development. However, it is not appropriate to model some real situations where stochastic fluctuations with long-range dependence might exist. Due to the long-range dependence of the fBm which was introduced by Hurst [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title=" Hurst, H: Long-term storage capacity in reservoirs. Trans. Am. Soc. Civ. Eng. 116, 400-410 (1951) " href="/articles/10.1186/s13662-016-0916-1#ref-CR5" id="ref-link-section-d76369347e348">5</a>], Kolmogorov [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title=" Kolmogorov, A: Wienersche spiralen und einige andere interessante kurven im Hilbertschen raum. C. R. (Dokl.) Acad. Sci. URSS 26, 115-118 (1940) " href="/articles/10.1186/s13662-016-0916-1#ref-CR6" id="ref-link-section-d76369347e351">6</a>], Mandelbrot [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title=" Mandelbrot, B, Van Ness, J: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422-427 (1968) " href="/articles/10.1186/s13662-016-0916-1#ref-CR7" id="ref-link-section-d76369347e354">7</a>] originally, SDEs driven by fBm have been used as the models of a number of practical problems in various fields, such as queueing theory, telecommunications, and economics [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title=" Chakravarti, N, Sebastian, K: Fractional Brownian motion models for polymers. Chem. Phys. Lett. 267, 9-13 (1997) " href="/articles/10.1186/s13662-016-0916-1#ref-CR8" id="ref-link-section-d76369347e358">8</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title=" Scheffer, R, Maciel, F: The fractional Brownian motion as a model for an industrial airlift reactor. Chem. Eng. Sci. 56, 707-711 (2001) " href="/articles/10.1186/s13662-016-0916-1#ref-CR10" id="ref-link-section-d76369347e361">10</a>].</p><p>On most occasions, the coefficients of SDEs driven by fBm are assumed to satisfy the Lipschitz condition. The existence and uniqueness of solutions of SDEs driven by fBm with Lipschitz condition have been studied by many scholars [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Lyons, T: Differential equations driven by rough signals. Rev. Mat. Iberoam. 14, 215-310 (1998) " href="/articles/10.1186/s13662-016-0916-1#ref-CR11" id="ref-link-section-d76369347e367">11</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title=" Biagini, F, Hu, Y, Oksendal, B, Zhang, T: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008) " href="/articles/10.1186/s13662-016-0916-1#ref-CR14" id="ref-link-section-d76369347e370">14</a>]. However, this Lipschitz condition seemed to be considerably strong when one discusses variable applications in real world. For example, the hybrid square root process and the one-dimensional semi-linear SDEs with Markov switching. Such models appear widely in many branches of science, engineering, industry and finance [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title=" Cox, J, Ingersoll, J, Ross, S: A theory of the term structure of interest rate. Econometrica 53, 385-407 (1985) " href="/articles/10.1186/s13662-016-0916-1#ref-CR15" id="ref-link-section-d76369347e373">15</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title=" Kwok, Y: Pricing multi-asset options with an external barrier. Int. J. Theor. Appl. Finance 1, 523-541 (1998) " href="/articles/10.1186/s13662-016-0916-1#ref-CR17" id="ref-link-section-d76369347e376">17</a>]. Therefore, it is important to obtain some weaker condition than the Lipschitz one under which the SDEs still have unique solutions. Fortunately, many researchers have investigated the SDEs under non-Lipschitz condition and they presented many meaningful results [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title=" Watanabe, S, Yamada, T: On the uniqueness of solution of stochastic differential equations II. J. Math. Kyoto Univ. 11(3), 553-563 (1971) " href="/articles/10.1186/s13662-016-0916-1#ref-CR18" id="ref-link-section-d76369347e379">18</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title=" Yamada, T, Watanabe, S: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155-167 (1971) " href="/articles/10.1186/s13662-016-0916-1#ref-CR22" id="ref-link-section-d76369347e383">22</a>]. But, to the best of our knowledge, the existence and uniqueness of solutions of SDEs driven by fBm with a non-Lipschitz condition have not been considered. Since fBm is neither a semi-martingale nor a Markov process, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality, which is crucial for SDEs driven by Brownian motion. Then it seems not to be very easy to obtain the existence and uniqueness of solutions to non-Lipschitz SDEs with fBm. This point motivates us to carry out the present study.</p><p>We in the present paper discuss the SDEs with fBm under the non-Lipschitz condition. Using the successive approximation method, the existence and uniqueness theorems of solutions to the following non-Lipschitz SDEs driven by fBm are proved: </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} X( t) = X( 0) + \int_{0}^{t} {b\bigl( {s,X( s)} \bigr)}\,ds + \int_{0}^{t} {\sigma\bigl( {s,X( s)} \bigr)} \,d{B^{H}}( s),\quad t \in[ {0,T} ], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (1.1) </div></div><p> where the initial data <span class="mathjax-tex">$X(0)=\xi$</span> is a random variable, <span class="mathjax-tex">$0< T<\infty$</span>, the process <span class="mathjax-tex">$B^{H}(t)$</span> represents the fBm with Hurst index <span class="mathjax-tex">$H\in(\frac {1}{2},1)$</span> defined in a complete probability space <span class="mathjax-tex">$(\Omega,\mathcal {F}, \mathbb{P})$</span>, and <span class="mathjax-tex">$b( {t,X( t)}):[0,T] \times R \to R$</span> and <span class="mathjax-tex">$\sigma( {t,X( t)}):[ {0,T} ] \times R \to R $</span> are all measurable functions; <span class="mathjax-tex">$\int_{0}^{t} \cdot{\,d{B^{H}}( s)}$</span> stands for the stochastic integral with respect to fBm.</p></div></div></section><section data-title="Preliminaries"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Preliminaries</h2><div class="c-article-section__content" id="Sec2-content"><p>Let <span class="mathjax-tex">$( {\Omega,\mathcal{F},\mathbb{P}})$</span> be a complete probability space. SDEs with respect to fBm have been interpreted via various stochastic integrals, such as the Wick integral, the Wiener integral, the Skorohod integral, and path-wise integrals [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title=" Mishura, Y: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008) " href="/articles/10.1186/s13662-016-0916-1#ref-CR13" id="ref-link-section-d76369347e949">13</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title=" Carmona, P, Coutin, L, Montseny, G: Stochastic integration with respect to fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 39, 27-68 (2003) " href="/articles/10.1186/s13662-016-0916-1#ref-CR23" id="ref-link-section-d76369347e952">23</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title=" Alòs, E, Nualart, D: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75(3), 129-152 (2003) " href="/articles/10.1186/s13662-016-0916-1#ref-CR26" id="ref-link-section-d76369347e955">26</a>]. In this paper, we consider the path-wise integrals [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title=" Russo, F, Vallois, P: Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97, 403-421 (1993) " href="/articles/10.1186/s13662-016-0916-1#ref-CR27" id="ref-link-section-d76369347e958">27</a>] with respect to fBm.</p><p>Let <span class="mathjax-tex">$\varphi:{{R}_{+} } \times{{R}_{+} } \to{{R}_{+} }$</span> be defined by </p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \varphi( {t,s}) = H( {2H - 1}){\vert {t - s} \vert ^{2H - 2}},\quad t,s \in{{R}_{+} }, \end{aligned}$$ </span></div></div><p> where <i>H</i> is a constant with <span class="mathjax-tex">$\frac{1}{2} < H < 1$</span>.</p><p>Let <span class="mathjax-tex">$g:{{R}_{+} } \to{R}$</span> be Borel measurable.</p><p>Define </p><div id="Equb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} L_{\varphi}^{2}( {{{R}_{+} }}) = \biggl\{ {g:\Vert g \Vert _{\varphi}^{2} = \int _{{{R}_{+} }} { \int_{{{R}_{+} }} {g( t)g( s)\varphi( {t,s})\,ds\,dt < \infty} } } \biggr\} . \end{aligned}$$ </span></div></div><p> If we equip <span class="mathjax-tex">$L^{2}_{\varphi}({R}_{+})$</span> with the inner product </p><div id="Equc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {\langle{{g_{1}},{g_{2}}} \rangle_{\varphi}} = \int_{{{R}_{+}}} { \int _{{{R}_{+}}} {{g_{1}}( t){g_{2}}( s) \varphi( {t,s})\,ds\,dt} },\quad {g_{1}},{g_{2}} \in L_{\varphi}^{2}( {{R_{+} }}), \end{aligned}$$ </span></div></div><p> then <span class="mathjax-tex">$L^{2}_{\varphi}({R}_{+})$</span> becomes a separable Hilbert space.</p><p>Let <span class="mathjax-tex">$\mathcal{S}$</span> be the set of smooth and cylindrical random variables of the form </p><div id="Equd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} F(\omega) = f \biggl( \int_{0}^{T}\psi_{1}(t)\,dB^{H}_{t}, \ldots, \int _{0}^{T}\psi_{n}(t)\,dB^{H}_{t} \biggr), \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$n \ge1$</span>, <span class="mathjax-tex">$f \in\mathcal{C}_{b}^{\infty}( {{{R}^{n}}})$</span> (<i>i.e.</i> <i>f</i> and all its partial derivatives are bounded), and <span class="mathjax-tex">${\psi_{i}} \in\mathcal{H}$</span>, <span class="mathjax-tex">$i = 1,2,\ldots, n$</span>. <span class="mathjax-tex">$\mathcal{H}$</span> is the completion of the measurable functions such that <span class="mathjax-tex">$\Vert \psi \Vert _{\varphi}^{2} <\infty$</span> and <span class="mathjax-tex">$\{\psi_{n}\}$</span> is a sequence in <span class="mathjax-tex">$\mathcal{H}$</span> such that <span class="mathjax-tex">$\langle \psi_{i}, \psi_{j}\rangle_{\varphi}=\delta_{ij}$</span>.</p><p>The Malliavin derivative <span class="mathjax-tex">$D_{t}^{H}$</span> of a smooth and cylindrical random variable <span class="mathjax-tex">$F\in\mathcal{S}$</span> is defined as the <span class="mathjax-tex">$\mathcal{H}$</span>-valued random variable: </p><div id="Eque" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} D_{t}^{H}F = \sum_{i = 1}^{n} {\frac{{\partial f}}{{\partial {x_{i}}}} \biggl( \int_{0}^{T}\psi_{1}(t)\,dB^{H}_{t}, \ldots, \int _{0}^{T}\psi_{n}(t)\,dB^{H}_{t} \biggr)} {\psi_{i}(t)}. \end{aligned}$$ </span></div></div> <p>Then, for any <span class="mathjax-tex">$p\geq1$</span>, the derivative operator <span class="mathjax-tex">$D_{t}^{H}$</span> is a closable operator from <span class="mathjax-tex">$L^{p}(\Omega)$</span> into <span class="mathjax-tex">$L^{p}(\Omega;\mathcal{H})$</span>. Next, we introduce the <i>φ</i>-derivative of <i>F</i>: </p><div id="Equf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} D_{t}^{\varphi}F = \int_{{{R}_{+} }} {\varphi( {t,v})} D_{v}^{H} F\,dv. \end{aligned}$$ </span></div></div> <p>The elements of <span class="mathjax-tex">$\mathcal{H}$</span> may not be functions but distributions of negative order. Thanks to this, it is convenient to introduce the space <span class="mathjax-tex">$\vert \mathcal{H} \vert $</span> of the measurable function <i>h</i> on <span class="mathjax-tex">$[ {0,T} ]$</span> satisfying </p><div id="Equg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \Vert h \Vert _{\vert \mathcal{H} \vert }^{2} = \int_{0}^{T} { \int_{0}^{T} {\bigl\vert {h( t)} \bigr\vert \bigl\vert {h( s)}\bigr\vert \varphi( {t,s})\,ds\,dt < \infty}}. \end{aligned}$$ </span></div></div><p> It is not difficult to show that <span class="mathjax-tex">$\vert \mathcal{H} \vert $</span> is a Banach space with the norm <span class="mathjax-tex">$\Vert {\cdot} \Vert _{\vert \mathcal{H} \vert }^{2}$</span>.</p><p>In addition, we denote by <span class="mathjax-tex">$D_{t}^{H,k}$</span> the iteration of the derivative operator for any integer <span class="mathjax-tex">$k\geq1$</span>. The Sobolev space <span class="mathjax-tex">${\mathbb {D}}^{k,p}$</span> is the closure of <span class="mathjax-tex">$\mathcal{S}$</span> with respect to the norm, for any <span class="mathjax-tex">$p\geq1$</span> (<span class="stix">⨂</span> denotes the tensor product), </p><div id="Equh" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \Vert F\Vert _{k,p}^{p}=\mathbb{E}\vert F\vert ^{p}+\mathbb{E}\sum_{j=1}^{k}{ \bigl\Vert D_{t}^{H,j}F\bigr\Vert _{\mathcal{H}^{\bigotimes j}}^{p}}. \end{aligned}$$ </span></div></div><p> Similarly, for a Hilbert space <i>U</i>, we denote by <span class="mathjax-tex">${\mathbb {D}}^{k,p}(U)$</span> the corresponding Sobolev space of <i>U</i>-valued random variables. For any <span class="mathjax-tex">$p>0$</span> we denote by <span class="mathjax-tex">${\mathbb{D}}^{1,p}(\vert \mathcal{H}\vert )$</span> the subspace of <span class="mathjax-tex">${\mathbb{D}}^{1,p}(\mathcal{H})$</span> formed by the elements <i>h</i> such that <span class="mathjax-tex">$h\in \vert \mathcal{H}\vert $</span>.</p><p>Biagini <i>et al.</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title=" Biagini, F, Hu, Y, Oksendal, B, Zhang, T: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008) " href="/articles/10.1186/s13662-016-0916-1#ref-CR14" id="ref-link-section-d76369347e3276">14</a>], Alos, Mazet and Nualart [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title=" Alòs, E, Mazet, O, Nualart, D: Stochastic calculus with respect to Gaussian process. Ann. Probab. 29, 766-801 (2001) " href="/articles/10.1186/s13662-016-0916-1#ref-CR24" id="ref-link-section-d76369347e3279">24</a>], Hu and Øksendal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title=" Hu, Y, Øksendal, B: Fractional white noise calculus and application to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, 1-32 (2003) " href="/articles/10.1186/s13662-016-0916-1#ref-CR9" id="ref-link-section-d76369347e3282">9</a>] have given more details as regards the fBm.</p> <h3 class="c-article__sub-heading" id="FPar1">Lemma 1</h3> <p> <i>Let</i> <span class="mathjax-tex">$u(t)$</span> <i>be a stochastic process in the space</i> <span class="mathjax-tex">${\mathbb{D}}^{1,2}(\vert \mathcal{H}\vert )$</span>, <i>satisfying</i> </p><div id="Equi" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \int_{0}^{T} { \int_{0}^{T} {\bigl\vert {D_{s}^{H}u( t)} \bigr\vert {{\vert {t - s} \vert }^{2H - 2}}\,ds\,dt} } < \infty, \end{aligned}$$ </span></div></div><p> <i>then the symmetric integral coincides with the forward and backward integrals</i> (<i>P</i>159,[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title=" Biagini, F, Hu, Y, Oksendal, B, Zhang, T: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008) " href="/articles/10.1186/s13662-016-0916-1#ref-CR14" id="ref-link-section-d76369347e3492">14</a>]).</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 2</h3> <p>The space <span class="mathjax-tex">$\mathcal{L}_{\varphi}[0,T]$</span> of integrands is defined as the family of stochastic processes <span class="mathjax-tex">$u(t)$</span> on <span class="mathjax-tex">$[0,T]$</span>, such that <span class="mathjax-tex">$\mathbb{E}\Vert {u( t)} \Vert _{\varphi}^{2} < \infty$</span>, <span class="mathjax-tex">$u(t)$</span> is <i>φ</i>-differentiable, the trace of <span class="mathjax-tex">$D_{s}^{\varphi}u( t)$</span> exists, <span class="mathjax-tex">$0 \le s \le T$</span>, <span class="mathjax-tex">$0 \le t \le T$</span>, and </p><div id="Equj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} \int_{0}^{T} { \int_{0}^{T} {{{\bigl[ {D_{t}^{\varphi}u( s)} \bigr]}^{2}}\,ds\,dt} } < \infty, \end{aligned}$$ </span></div></div><p> and for each sequence of partitions <span class="mathjax-tex">$( {{\pi_{n}},n \in\mathbb{N}})$</span> such that <span class="mathjax-tex">$\vert {{\pi_{n}}} \vert \to0$</span> as <span class="mathjax-tex">$n \to\infty$</span>, </p><div id="Equk" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sum_{i = 0}^{n - 1} {\mathbb{E} \biggl[ { \int_{t_{i}^{( n)}}^{t_{i + 1}^{( n)}} { \int_{t_{j}^{( n)}}^{t_{j + 1}^{( n)}} {\bigl\vert {D_{s}^{\varphi}u^{\pi}\bigl(t_{i}^{( n)}\bigr) D_{t}^{\varphi}u^{\pi}\bigl(t_{j}^{( n)}\bigr) -D_{s}^{\varphi}{u(t)}D_{t}^{\varphi}{u(s)}} \bigr\vert \,ds\,dt} } } \biggr]} \end{aligned}$$ </span></div></div><p> and </p><div id="Equl" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E}\bigl[ {\bigl\Vert {{u^{\pi}} - u} \bigr\Vert _{\varphi}^{2}} \bigr] \end{aligned}$$ </span></div></div><p> tend to 0 as <span class="mathjax-tex">$n\to\infty$</span>, where <span class="mathjax-tex">${\pi_{n}} = t_{0}^{( n)} < t_{1}^{( n)} < \cdots < t_{n - 1}^{( n)} < t_{n}^{( n)} = T$</span>.</p> <h3 class="c-article__sub-heading" id="FPar3">Lemma 3</h3> <p> <i>Let</i> <span class="mathjax-tex">$B^{H}(t)$</span> <i>be a fBm with</i> <span class="mathjax-tex">$\frac{1}{2}< H<1$</span>, <i>and</i> <span class="mathjax-tex">$u(t)$</span> <i>be a stochastic process in</i> <span class="mathjax-tex">${{\mathbb{D}}^{1,2}}( {\vert \mathcal{H} \vert }) \cap {\mathcal{L}_{\varphi}}[ {0,T} ]$</span>, <i>then for every</i> <span class="mathjax-tex">$T<\infty$</span>, </p><div id="Equm" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} { \biggl[ { \int_{0}^{T} {u( s)\,{d^{\circ}} {B^{H}}( s)} } \biggr]^{2}} \le2H{T^{2H - 1}} \mathbb{E} \biggl[ { \int_{0}^{T} {{{ \bigl\vert {u( s)}\bigr\vert }^{2}}\,ds} } \biggr] + 4T\mathbb{E} \int_{0}^{T} \bigl[{D_{s}^{\varphi}u( s)\bigr]^{2}\,ds}. \end{aligned}$$ </span></div></div><p> <i>The detailed proof of Lemma </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar3">3</a> <i>can be found in the authors’ previous work</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title=" Xu, Y, Pei, B, Guo, R: Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst., Ser. B 20, 2257-2267 (2015) " href="/articles/10.1186/s13662-016-0916-1#ref-CR28" id="ref-link-section-d76369347e4897">28</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title=" Xu, Y, Pei, B, Li, Y: An averaging principle for stochastic differential delay equations with fractional Brownian motion. Abstr. Appl. Anal. 2014, Article ID 479195 (2014) " href="/articles/10.1186/s13662-016-0916-1#ref-CR30" id="ref-link-section-d76369347e4900">30</a>].</p> <p>In this paper, we always assume the following non-Lipschitz condition, which was proposed by Yamada and Watanabe [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title=" Yamada, T, Watanabe, S: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155-167 (1971) " href="/articles/10.1186/s13662-016-0916-1#ref-CR22" id="ref-link-section-d76369347e4907">22</a>], is satisfied.</p> <h3 class="c-article__sub-heading" id="FPar4">Hypothesis 4</h3> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>There exists a function <span class="mathjax-tex">$\kappa( q)>0$</span>, <span class="mathjax-tex">$q > 0$</span>, <span class="mathjax-tex">$\kappa( 0 )=0$</span> such that <span class="mathjax-tex">$\kappa( q)$</span> is a continuous non-decreasing, concave function and <span class="mathjax-tex">$\int_{0 + } {\frac{{dq}}{{\kappa( q)}}} = + \infty$</span>,</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p> <span class="mathjax-tex">$b( {t,0})$</span>, <span class="mathjax-tex">$\sigma( {t,0})$</span> are locally integral with respect to <i>t</i>,</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>Furthermore, <span class="mathjax-tex">$\forall t \in[ {0,T} ]$</span>, <span class="mathjax-tex">$b( {t,\cdotp}),\sigma( {t,\cdotp}) \in {\mathcal{L}_{\varphi}}[ {0,T} ] \cap{\mathbb{D}^{1,2}}( {\vert \mathcal{H} \vert } )$</span>, we have </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \mathbb{E} {\bigl\vert {b( {t,X}) - b( {t,Y})} \bigr\vert ^{2}} + \mathbb{E} {\bigl\vert {\sigma( {t,X}) - \sigma( {t,Y})} \bigr\vert ^{2}} \\& \quad {} + \mathbb{E} {\bigl\vert {D_{t}^{\varphi}\bigl( {\sigma({t,X}) - \sigma( {t,Y})} \bigr)} \bigr\vert ^{2}} \le\kappa\bigl( {\mathbb{E} {{\vert {X - Y} \vert }^{2}}} \bigr). \end{aligned}$$ </span></div><div class="c-article-equation__number"> (2.1) </div></div> </li> </ol> <p>The above-mentioned Hypothesis <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a> is the so-called non-Lipschitz condition. The non-Lipschitz condition has a variety of forms [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title=" Albeverio, S, Brzézniak, Z, Wu, J: Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371, 309-322 (2010) " href="/articles/10.1186/s13662-016-0916-1#ref-CR31" id="ref-link-section-d76369347e5576">31</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title=" Xu, Y, Pei, B, Wu, J: Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. Stoch. Dyn. (2016). doi: 10.1142/S0219493717500137 " href="/articles/10.1186/s13662-016-0916-1#ref-CR34" id="ref-link-section-d76369347e5579">34</a>]. Here, we consider one kind of them. In particular, we see clearly that if we let <span class="mathjax-tex">$\kappa( q) = K'q$</span>, then the non-Lipschitz condition reduces to the Lipschitz condition. In other words, the non-Lipschitz condition is weaker than the Lipschitz condition.</p><p>Now, we give some concrete examples of the function <i>κ</i>. Let <span class="mathjax-tex">$K'>0$</span> and let <span class="mathjax-tex">$\mu\in\mathopen{]}0,1[$</span> be sufficiently small. Define </p><div id="Equn" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& {\kappa_{1}} ( x) = K'x,\quad x \ge0,\\& {\kappa_{2}} ( x) = \textstyle\begin{cases} x\log({x^{ - 1}}),& 0 \le x \le\mu, \\ \mu\log({\mu^{ - 1}}) + \kappa'_{2} ( {\mu-}) ( {x - \mu}),& x > \mu, \end{cases}\displaystyle \\& {\kappa_{3}} ( x) = \textstyle\begin{cases} x\log({x^{ - 1}})\log\log({x^{ - 1}}),& 0 \le x \le\mu,\\ \mu\log({\mu^{ - 1}})\log\log({\mu^{-1}}) + \kappa'_{3} ( {\mu - }) ( {x - \mu}),& x > \mu, \end{cases}\displaystyle \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$\kappa'$</span> denotes the derivative of the function <i>κ</i>. They are all concave and non-decreasing functions satisfying <span class="mathjax-tex">$\int_{0 + } {\frac{1}{{{\kappa_{i}}( { x})}}}\,dx = \infty$</span> (<span class="mathjax-tex">$i = 1,2,3$</span>).</p></div></div></section><section data-title="The main theorems"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>The main theorems</h2><div class="c-article-section__content" id="Sec3-content"><p>In this section, using an iteration of the Picard type, we will discuss the solutions for non-Lipschitz SDEs with fBm. Let <span class="mathjax-tex">${X_{0}}( t) \equiv \xi$</span> be a random variable with <span class="mathjax-tex">$\mathbb{E}{\vert \xi \vert ^{2}} < + \infty$</span>, and construct an approximate sequence of stochastic process <span class="mathjax-tex">$\{ X_{k}(t)\}_{k \geq1}$</span> as follows: </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {X_{k}}( t) = \xi + \int_{0}^{t} {b\bigl( {s,{X_{k - 1}}( s)} \bigr)}\,ds + \int _{0}^{t} {\sigma\bigl( {s,{X_{k - 1}}( s)} \bigr)} \,d^{\circ}{B^{H}}( s),\quad k = 1,2, \ldots. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.1) </div></div><p> Hereafter, we assume that <span class="mathjax-tex">$1 \le T < + \infty$</span> without losing generality.</p><p>First, we given the following four key lemmas. The proofs for Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar5">5</a> and Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar6">6</a> will be presented in the Appendix.</p> <h3 class="c-article__sub-heading" id="FPar5">Lemma 5</h3> <p> <i>There exists a positive number</i> <i>K</i>, <span class="mathjax-tex">$\forall b( {t,\cdot}),\sigma( {t,\cdot}) \in{\mathcal{L}_{\varphi}}[ {0,T} ] \cap{\mathbb{D}^{1,2}}( {\vert \mathcal{H} \vert })$</span>, <span class="mathjax-tex">$t \in[ {0,T} ]$</span>, <i>and we have</i> </p><div id="Equo" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {b( {t,X})} \bigr\vert ^{2}} + \mathbb{E} {\bigl\vert {\sigma( {t,X})} \bigr\vert ^{2}} + \mathbb{E} {\bigl\vert {D_{t}^{\varphi}\sigma( {t,X})} \bigr\vert ^{2}} \le K\bigl( {1 + \mathbb {E} {{\vert X \vert }^{2}}} \bigr). \end{aligned}$$ </span></div></div> <h3 class="c-article__sub-heading" id="FPar6">Lemma 6</h3> <p> <i>Under the conclusion of Lemma </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar5">5</a>, <i>one can get</i> </p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{k}}( t)} \bigr\vert ^{2}} \le{C_{1}},\quad k = 1,2,\ldots,t \in[0,T], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.2) </div></div><p> <i>where</i> <span class="mathjax-tex">${C_{1}} = 3( {1 + \mathbb{E}{{\vert \xi \vert }^{2}}})\exp( {12K{T^{2}}} )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar7">Lemma 7</h3> <p> <i>If</i> <span class="mathjax-tex">$b(t,X)$</span> <i>and</i> <span class="mathjax-tex">$\sigma(t, X)$</span> <i>satisfy the Hypothesis </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a>, <i>then for</i> <span class="mathjax-tex">$t \in[0,T]$</span>, <span class="mathjax-tex">$n \ge1$</span>, <span class="mathjax-tex">$k \geq1$</span>, <i>we have</i> </p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{n}}( s)} \bigr\vert ^{2}} \le{C_{2}} \int_{0}^{t} {\kappa\bigl( {\mathbb{E} {{\bigl\vert {{X_{n + k - 1}}( s) - X_{n - 1}( s)}\bigr\vert }^{2}}} \bigr)}\,ds \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.3) </div></div><p> <i>and</i> </p><div id="Equp" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{n+k}}( s) -{X_{n}}( s)} \bigr\vert ^{2}} \le{C_{3}}t, \end{aligned}$$ </span></div></div><p> <i>where</i> <span class="mathjax-tex">$C_{2}=8T$</span> <i>and</i> <span class="mathjax-tex">$C_{3}$</span> <i>is a constant</i>.</p> <h3 class="c-article__sub-heading" id="FPar8">Proof</h3> <p>For <span class="mathjax-tex">$0 \le s \le t$</span>, we show that </p><div id="Equq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \mathbb{E}\bigl\vert X_{n+k}( s) - X_{n}( s)\bigr\vert ^{2} \\& \quad \le2\mathbb{E}\biggl\vert \int_{0}^{s} \bigl(b\bigl( s_{1},X_{n+k - 1}(s_{1}) \bigr) - b\bigl( s_{1},X_{n - 1}(s_{1}) \bigr) \bigr)\,d{s_{1}} \biggr\vert ^{2} \\& \qquad {} + 2\mathbb{E}\biggl\vert \int_{0}^{s} \bigl(\sigma\bigl( s_{1},X_{n+k - 1}(s_{1}) \bigr) - \sigma\bigl( s_{1},X_{n - 1}(s_{1}) \bigr) \bigr) \,d^{\circ}{B^{H}}(s_{1}) \biggr\vert ^{2} \\& \quad \le8T\mathbb{E} \int_{0}^{t} \bigl[\bigl\vert b\bigl( s_{1},X_{n+k - 1}(s_{1}) \bigr) - b(s_{1},X_{n - 1}(s_{1}) \bigr\vert ^{2} \\& \qquad {} +\bigl\vert \sigma\bigl( s_{1},X_{n+k - 1}(s_{1}) \bigr) - \sigma( s_{1},X_{n -1}(s_{1})\bigr\vert ^{2} \\& \qquad {}+\bigl\vert D_{s_{1}}^{\varphi}\bigl(\sigma \bigl( s_{1},X_{n+k - 1}(s_{1}) \bigr) - \sigma \bigl(s_{1},X_{n - 1}(s_{1})\bigr)\bigr)\bigr\vert ^{2}\bigr]\,d{s_{1}} \\& \quad \le{C_{2}} \int_{0}^{t} {\kappa\bigl( {\mathbb{E} {{\bigl\vert {{X_{n + k - 1}}( s) -X_{n - 1}( s)} \bigr\vert }^{2}}} \bigr)}\,ds. \end{aligned}$$ </span></div></div><p> Then it is easy to verify </p><div id="Equr" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{n+k}}( s) -{X_{n}}( s)} \bigr\vert ^{2}} \le& {C_{2}} \int_{0}^{t} {\kappa\bigl( {\mathbb{E} {{ \bigl\vert {{X_{n+ k - 1}}( s) - X_{n - 1}( s)} \bigr\vert }^{2}}} \bigr)}\,ds \\ \le& {C_{2}} \int_{0}^{t} {\kappa( {4{C_{1}}})}\,ds \le{C_{3}}t. \end{aligned}$$ </span></div></div><p> This completes the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar7">7</a>. □</p> <p>Now, choose <span class="mathjax-tex">$0 < {T_{1}} \le T$</span>, such that <span class="mathjax-tex">$t \in[ {0,{T_{1}}} ]$</span>, for <span class="mathjax-tex">${\kappa_{1}}( {{C_{3}}t}) \le{C_{3}}$</span>, <span class="mathjax-tex">${\kappa_{1}}( q) = {C_{2}}\kappa( q)$</span> holds. We should note that in the following part, we first of all prove the following main theorem, Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar11">9</a>, in the time interval <span class="mathjax-tex">$[0,{T_{1}}]$</span>, then we extend the result in the whole interval <span class="mathjax-tex">$[0,T]$</span>. Fix <span class="mathjax-tex">$k \geq1$</span> arbitrarily and define two sequences of functions <span class="mathjax-tex">${\{ {{\phi_{n}}( t)} \}_{n = 1,2, \ldots}}$</span> and <span class="mathjax-tex">${\{ {{{\tilde{\phi}}_{n,k}}( t)} \}_{n = 1,2, \ldots}}$</span>, where </p><div id="Equs" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& {\phi_{1}}( t)= {C_{3}}t, \\& {\phi_{n + 1}}( t) = \int_{0}^{t} {{\kappa_{1}}\bigl( {{ \phi_{n}}( s)} \bigr)}\,ds, \\& {\tilde{\phi}_{n,k}}( t) = \sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{n}}( s)} \bigr\vert ^{2}},\quad n = 1,2, \ldots. \end{aligned}$$ </span></div></div> <h3 class="c-article__sub-heading" id="FPar9">Lemma 8</h3> <p> <i>Under the Hypothesis </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a>, </p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} 0 \le{\tilde{\phi}_{n,k}}( t) \le{\phi_{n}}( t) \le{\phi_{n - 1}}( t) \le \cdots \le{\phi_{1}}( t),\quad t \in[ {0,{T_{1}}} ], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.4) </div></div><p> <i>for all positive integer</i> <i>n</i>.</p> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar7">7</a>, we have </p><div id="Equt" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {\tilde{\phi}_{1,k}}( t) = \sup _{0 \le s \le t} \mathbb{E} { \bigl\vert {{X_{1+k}}( s) - {X_{1}}( s)} \bigr\vert ^{2}} \le{C_{3}}t = {\phi _{1}}( t),\quad t \in[ {0,{T_{1}}} ]. \end{aligned}$$ </span></div></div><p> Then, since <span class="mathjax-tex">${\kappa_{1}}( q) = {C_{2}}\kappa( q)$</span>, <span class="mathjax-tex">$\kappa( q) $</span> is a concave function and </p><div id="Equu" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{k + 1}}( s) - {X_{1}}( s)} \bigr\vert ^{2}} \le\sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{k + 1}}( s) - {X_{1}}( s)}\bigr\vert ^{2}} = {\tilde{\phi}_{1,k}}( t),\quad 0 \le s \le t, \end{aligned}$$ </span></div></div><p> it is easy to verify </p><div id="Equv" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {{\tilde{\phi}}_{2,k}}( t) =& \sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{2 + k}}( s) - {X_{2}}( s)} \bigr\vert ^{2}} \\ \le& {C_{2}} \int_{0}^{t} {\kappa\bigl( {\mathbb{E} {{\bigl\vert {{X_{k + 1}}( s) -{X_{1}}( s)} \bigr\vert }^{2}}} \bigr)}\,ds \\ \le& \int_{0}^{t} {{\kappa_{1}}\bigl( {{{ \tilde{\phi}}_{1,k}}( s)} \bigr)}\,ds \le \int_{0}^{t} {{\kappa_{1}}\bigl( {{ \phi_{1}}( s)} \bigr)}\,ds \\ =& {\phi_{2}}( t) = \int_{0}^{t} {{\kappa_{1}}( {{C_{3}}s})}\,ds \\ \le& {C_{3}}t = { \phi_{1}}( t),\quad t \in[ {0,{T_{1}}} ]. \end{aligned}$$ </span></div></div><p> That is to say, for <span class="mathjax-tex">$n=2$</span>, we have </p><div id="Equw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {\tilde{\phi}_{2,k}}( t) \le{\phi_{2}}( t) \le{ \phi_{1}}( t),\quad t \in[ {0,{T_{1}}} ]. \end{aligned}$$ </span></div></div><p> Next, assume (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ6">3.4</a>) for <span class="mathjax-tex">$n \geq2$</span> and by the assumption for <i>n</i> </p><div id="Equx" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{n}}( s)} \bigr\vert ^{2}} \le\sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{n}}( s)}\bigr\vert ^{2}} = {\tilde{\phi}_{n,k}}( t) \le{\phi_{n}}( t), \end{aligned}$$ </span></div></div><p> it is easy to verify for <span class="mathjax-tex">$n+1$</span> </p><div id="Equy" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {{\tilde{\phi}}_{n + 1,k}}( t) =& \sup _{0 \le s \le t} \mathbb{E} {\bigl\vert {{X_{n + k + 1}}( s) - {X_{n + 1}}( s)} \bigr\vert ^{2}} \\ \le& \int_{0}^{t} {{\kappa_{1}}\bigl( { \mathbb{E} {{\bigl\vert {{X_{n + k}}( s) -{X_{n}}( s)} \bigr\vert }^{2}}} \bigr)}\,ds \\ \le& \int_{0}^{t} {{\kappa_{1}}\bigl( {{{ \tilde{\phi}}_{n,k}}( s)} \bigr)}\,ds \\ \le& \int_{0}^{t} {{\kappa_{1}}\bigl( {{ \phi_{n}}( s)} \bigr)}\,ds = {\phi_{n + 1}}( t) \\ \le& \int_{0}^{t} {{\kappa_{1}}\bigl( {{ \phi_{n - 1}}( s)} \bigr)}\,ds = {\phi _{n}}( t),\quad t \in[ {0,{T_{1}}} ]. \end{aligned}$$ </span></div></div><p> This completes the proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar9">8</a>. □</p> <h3 class="c-article__sub-heading" id="FPar11">Theorem 9</h3> <p> <i>Under the Hypothesis </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a>, <i>then</i> </p><div id="Equz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le T} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div> <p>By Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar11">9</a>, we say that <span class="mathjax-tex">$\{{X_{k}}(\cdotp)\}_{k\geq1}$</span> is a Cauchy sequence and define its limit as <span class="mathjax-tex">$X( \cdotp)$</span>. Then letting <span class="mathjax-tex">$k\to \infty$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ3">3.1</a>), we finally see that the solutions to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ1">1.1</a>) exist.</p> <h3 class="c-article__sub-heading" id="FPar12">Proof</h3> <p> <i>Step</i> 1: In this step we shall show </p><div id="Equaa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le{T_{1}}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div><p> By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar9">8</a>, we know <span class="mathjax-tex">${\phi_{n}}( t)$</span> decreases monotonically when <span class="mathjax-tex">$n \to\infty$</span> and <span class="mathjax-tex">${\phi_{n}}( t)$</span> is non-negative function on <span class="mathjax-tex">$t \in[ {0,{T_{1}}} ]$</span>. Therefore, we can define the limit function <span class="mathjax-tex">$\phi( t)$</span> by <span class="mathjax-tex">${\phi_{n}}( t) \downarrow\phi( t)$</span>. It is easy to verify that <span class="mathjax-tex">$\phi( 0) = 0$</span> and <span class="mathjax-tex">$\phi( t)$</span> is a continuous function on <span class="mathjax-tex">$t \in[ {0,{T_{1}}} ]$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title=" Xu, Y, Pei, B, Guo, G: Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise. Appl. Math. Comput. 263, 398-409 (2015) " href="/articles/10.1186/s13662-016-0916-1#ref-CR35" id="ref-link-section-d76369347e12291">35</a>]. According to the definition of <span class="mathjax-tex">${\phi_{n}}( t)$</span> and <span class="mathjax-tex">${\phi}( t)$</span>, we obtain </p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \phi( t) = \lim _{n \to\infty} {\phi_{n + 1}}( t) = \lim _{n \to\infty} \int_{0}^{t} {{\kappa_{1}}\bigl( {{ \phi_{n}}( s)} \bigr)}\,ds = \int_{0}^{t} {{\kappa_{1}}\bigl( {\phi( s)} \bigr)}\,ds,\quad t \in[ {0,{T_{1}}} ]. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.5) </div></div><p> Since <span class="mathjax-tex">$\phi( 0) = 0$</span> and </p><div id="Equab" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \int_{0 + } {\frac{{dq}}{{{\kappa_{1}}( q)}}} = \frac {1}{{{C_{2}}}} \int_{0 + } {\frac{{dq}}{{\kappa( q)}}} = + \infty, \end{aligned}$$ </span></div></div><p> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ7">3.5</a>) implies <span class="mathjax-tex">$\phi( t) \equiv0$</span>, <span class="mathjax-tex">$t\in[0,T_{1}]$</span>.</p> <p>Therefore we obtain </p><div id="Equac" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} 0 \le\lim _{k,n \to\infty} \sup _{0 \le t \le{T_{1}}} \mathbb{E} { \bigl\vert {{X_{n + k}}( t) - {X_{n}}( t)} \bigr\vert ^{2}} = \lim _{k,n \to\infty} {\tilde{\phi}_{n,k}}( {{T_{1}}}) \le\lim _{n \to\infty} {\phi_{n}}( {{T_{1}}}) = 0, \end{aligned}$$ </span></div></div><p> namely, </p><div id="Equad" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le{T_{1}}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div> <p> <i>Step</i> 2: Define </p><div id="Equae" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} T_{2} = \sup \Bigl\{ {\tilde{T} :\tilde{T} \in[ {0,T} ] \mbox{ and } \lim _{n,i \to\infty} \sup _{0 \le t \le\tilde{T}} \mathbb{E} \bigl\vert {X_{n}}( t) - {X_{i}}( t) \bigr\vert ^{2}= 0} \Bigr\} . \end{aligned}$$ </span></div></div><p> Immediately, we can observe <span class="mathjax-tex">$0 < {T_{1}} \le T_{2} \le T$</span>. Now, we shall show </p><div id="Equaf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div><p> Let <span class="mathjax-tex">$\varepsilon>0$</span> be an arbitrary positive number. Choose <span class="mathjax-tex">$S_{0}$</span> so that <span class="mathjax-tex">$0 < {S_{0}} < \min( {T_{2},1})$</span>. And </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {C_{4}} {S_{0}} < \frac{\varepsilon}{{10}}, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.6) </div></div><p> where <span class="mathjax-tex">${C_{4}} = 8K({1 + {K_{1}}( {1 + \mathbb{E}{{\vert \xi \vert }^{2}}} )}){S_{0}}$</span>.</p> <p>From the definition of <span class="mathjax-tex">$T_{2}$</span>, we have </p><div id="Equag" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le T_{2} - {S_{0}}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div><p> Then, for large enough <i>N</i>, we observe </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{0 \le t \le T_{2} - {S_{0}}} \mathbb{E} {\bigl\vert {{X_{n}}(t) - {X_{i}}( t)} \bigr\vert ^{2}} < \frac{\varepsilon}{{10}},\quad n,i \ge N. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.7) </div></div><p> On the other hand, one can get </p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{T_{2} - {S_{0}} \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) -{X_{i}}( t)} \bigr\vert ^{2}} \le& 3\sup _{T_{2} - {S_{0}} \le t \le T_{2}} \mathbb {E} {\bigl\vert {{X_{n}}( t) - {X_{n}}( {T_{2} - {S_{0}}})} \bigr\vert ^{2}} \\ &{}+3\mathbb{E} {\bigl\vert {{X_{n}}( {T_{2} - {S_{0}}}) - {X_{i}}( {T_{2} - {S_{0}}})}\bigr\vert ^{2}} \\ &{}+3\sup _{T_{2} - {S_{0}} \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{i}}( {T_{2} - {S_{0}}}) - {X_{i}}( t)} \bigr\vert ^{2}} \\ =& 3{I_{1}} + 3{I_{2}} + 3{I_{3}}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.8) </div></div><p> Now, using Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar3">3</a>, we obtain </p><div id="Equah" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {I_{1}} =& \sup _{T_{2} - {S_{0}} \le t \le T_{2}} \mathbb {E} {\bigl\vert {{X_{n}}( t) - {X_{n}}( {T_{2} - {S_{0}}})} \bigr\vert ^{2}} \\ \le& 2{S_{0}} \mathbb{E} \int_{T_{2} - {S_{0}}}^{T_{2}} {{{\bigl\vert {b\bigl( s_{1},X_{n -1}(s_{1}) \bigr)} \bigr\vert }^{2}}} \,d{s_{1}} \\ &{}+ 4H{S_{0}}^{2H - 1} \mathbb{E} \int_{T_{2} - {S_{0}}}^{T_{2}} {{{\bigl\vert {\sigma \bigl(s_{1},X_{n - 1}(s_{1}) \bigr)} \bigr\vert }^{2}}} \,d{s_{1}} \\ &{}+ 8{S_{0}}\mathbb{E} \int_{T_{2} - {S_{0}}}^{T_{2}} {{{\bigl\vert {D_{s_{1}}^{\varphi}\sigma\bigl( s_{1},X_{n - 1}(s_{1}) \bigr)} \bigr\vert }^{2}}} \,d{s_{1}} \\ \le& 8{S_{0}} \int_{T_{2} - {S_{0}}}^{T_{2}} {K\bigl({1 + {K_{1}}\bigl( {1 + \mathbb {E} {{\vert \xi \vert }^{2}}} \bigr)} \bigr)} \,d{s_{1}} \\ \le& 8S_{0}^{2}K\bigl( {1 + {K_{1}}\bigl( {1 + \mathbb{E} {{\vert \xi \vert }^{2}}} \bigr)} \bigr). \end{aligned}$$ </span></div></div><p> Therefore by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ8">3.6</a>) we have </p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {I_{1}} \le\frac{\varepsilon}{{10}} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.9) </div></div><p> and </p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {I_{3}} \le\frac{\varepsilon}{{10}}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.10) </div></div><p> Meanwhile, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ9">3.7</a>) implies </p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {I_{2}} = \mathbb{E} {\bigl\vert {{X_{n}}( {T_{2} - {S_{0}}}) - {X_{i}}( {T_{2} - {S_{0}}})}\bigr\vert ^{2}} < \frac{\varepsilon}{{10}},\quad n,i \ge N. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.11) </div></div><p> Now putting (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ9">3.7</a>)-(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ13">3.11</a>) together, we have </p><div id="Equai" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{0 \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) -{X_{i}}( t)} \bigr\vert ^{2}} \le& \sup _{0 \le t \le T_{2} - {S_{0}}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} \\ &{}+ \sup _{T_{2} - {S_{0}} \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} \\ \le& \frac{\varepsilon}{{10}} + 3{I_{1}} + 3{I_{2}} + 3{I_{3}} < \varepsilon. \end{aligned}$$ </span></div></div><p> That is to say, </p><div id="Equaj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\lim _{n,i \to\infty} \sup _{0 \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. $$</span></div></div> <p> <i>Step</i> 3: Using the method of reduction to absurdity, we shall show <span class="mathjax-tex">$T_{2}=T$</span>. Assume <span class="mathjax-tex">$T_{2}< T$</span>, we can choose a sequence of numbers <span class="mathjax-tex">${\{ {{a_{i}}} \} _{i = 1,2, \ldots}}$</span> so that <span class="mathjax-tex">${a_{i}} \downarrow0$</span> (<span class="mathjax-tex">${i \to + \infty }$</span>) and for <span class="mathjax-tex">$n > i \ge1$</span>, </p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{0 \le t \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n}}( t) -{X_{i}}( t)} \bigr\vert ^{2}} \le{a_{i}}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.12) </div></div><p> We shall divide the step into several sub-steps.</p> <p>First, for <span class="mathjax-tex">$n > i \ge1$</span>, we shall show </p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\bigl\vert {{X_{n}}( s) - {X_{i}}( s)} \bigr\vert ^{2}} \le3{a_{i}} + {C_{5}}t,\quad T_{2} + t \le T, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.13) </div></div><p> where <span class="mathjax-tex">${C_{5}} = 12TK({1 + {K_{1}}( {1 + \mathbb{E}{{\vert \xi \vert }^{2}}})})$</span>.</p> <p>To show this, set </p><div id="Equak" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& J_{1}^{( i)} = \mathbb{E} {\bigl\vert {{X_{n}}( {T_{2}}) - {X_{i}}( {T_{2}})} \bigr\vert ^{2}}, \\& J_{2}^{( i)}( t) = \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\biggl\vert { \int_{T_{2}}^{s} {\bigl( {b\bigl( s_{1},X_{n - 1}(s_{1}) \bigr) - b\bigl( {{s_{1}},{X_{i - 1}}(s_{1})} \bigr)} \bigr)\,d{s_{1}}} } \biggr\vert ^{2}}, \\& J_{3}^{( i)}( t)= \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\biggl\vert { \int_{T_{2}}^{s} {\bigl( {\sigma\bigl( s_{1},X_{n - 1}(s_{1}) \bigr) - \sigma\bigl( {{s_{1}},{X_{i - 1}}(s_{1})} \bigr)} \bigr)} \,d^{\circ}{B^{H}}(s_{1})} \biggr\vert ^{2}}. \end{aligned}$$ </span></div></div><p> Then (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ14">3.12</a>) implies <span class="mathjax-tex">$J_{1}^{( i)} \le{a_{i}}$</span> and </p><div id="Equal" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} J_{2}^{i}( t) + J_{3}^{i}( t) \le& 4T \mathbb{E} \int_{T_{2}}^{T_{2}+ t} \bigl[ \bigl\vert b \bigl(s_{1},X_{n - 1}(s_{1})\bigr) - b \bigl(s_{1},X_{i - 1}(s_{1})\bigr) \bigr\vert ^{2} \\ &{}+\bigl\vert \sigma\bigl(s_{1},X_{n - 1}(s_{1}) \bigr) - \sigma\bigl(s_{1},X_{i - 1}(s_{1})\bigr) \bigr\vert ^{2} \\ &{}+ \bigl\vert D_{s_{1}}^{\varphi}\bigl(\sigma\bigl(s_{1},X_{n - 1}(s_{1})\bigr) - \sigma\bigl(s_{1},X_{i -1}(s_{1})\bigr) \bigr)\bigr\vert ^{2}\bigr]\,ds_{1} \\ \le& 4TK\bigl(1 + {K_{1}}\bigl( 1 + \mathbb{E}\vert \xi \vert ^{2}\bigr) \bigr)t. \end{aligned}$$ </span></div></div><p> Therefore </p><div id="Equam" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\bigl\vert {{X_{n}}( s) - {X_{i}}( s)} \bigr\vert ^{2}} \le& 3J_{1}^{( i)} + 3J_{2}^{( i)}( t) + 3J_{3}^{( i)}( t) \\ \le& 3{a_{i}} + {C_{5}}t,\quad T_{2} + t \le T. \end{aligned}$$ </span></div></div> <p>Next, we shall show an assertion which is analogous to Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar9">8</a>. To state the assertion, we need to introduce several notations.</p> <p>Choose a positive number <span class="mathjax-tex">$0 < \eta \le T - T_{2}$</span> and a positive integer <span class="mathjax-tex">$j \geq1$</span>, so that </p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {C_{6}}\kappa( {3{a_{j}} + {C_{5}}t}) \le{C_{5}},\quad t \in[ {0,\eta} ],{\kappa _{2}}( q) = {C_{6}}\kappa( q), \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.14) </div></div><p> where <span class="mathjax-tex">$C_{6}=12T$</span>.</p> <p>Introduce the sequence of functions <span class="mathjax-tex">${\{ {{\psi_{k}}( t)} \}_{k = 1,2, \ldots}}$</span>, <span class="mathjax-tex">$t \in[ {0,\eta} ]$</span>, defined by </p><div id="Equan" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& {\psi_{1}}( t) = 3{a_{j}} + {C_{5}}t, \\ & { \psi_{k + 1}}( t)= 3{a_{j + k}} + \int_{0}^{t} {{\kappa_{2}}\bigl( {{\psi _{k}}( s)} \bigr)\,ds} , \\& {\tilde{\psi}_{k,n}}( t)= \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{j + k}}( s)} \bigr\vert ^{2}}. \end{aligned}$$ </span></div></div><p> Now, the assertion to be proved is the following: </p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {\tilde{\psi}_{k,n}}( t) \le{\psi_{k}}( t) \le{\psi_{k - 1}}( t) \le \cdots \le{\psi_{1}}( t),\quad t \in[ {0, \eta} ], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.15) </div></div><p> for all positive integer <i>k</i>.</p> <p>Noticing that <span class="mathjax-tex">${\kappa_{2}}( q)$</span> is a non-decreasing, concave function, and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ15">3.13</a>) holds, from this for <span class="mathjax-tex">$k=1$</span>, we work out </p><div id="Equao" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {{\tilde{\psi}}_{1,n}}( t) =& \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\bigl\vert {{X_{n + 1}}( s) - {X_{j + 1}}( s)} \bigr\vert ^{2}} \\ \le& 3a_{j + 1} + {C_{6}} \mathbb{E} \int_{T_{2}}^{T_{2} + t} \bigl[ \bigl\vert b \bigl({s_{1}},X_{n}(s_{1})\bigr) - b\bigl( s_{1},X_{j}(s_{1}) \bigr) \bigr\vert ^{2} \\ & {}+ \bigl\vert \sigma\bigl( s_{1},X_{n}(s_{1}) \bigr) - \sigma\bigl(s_{1},X_{j}(s_{1})\bigr) \bigr\vert ^{2} \\ &{}+\bigl\vert D_{s_{1}}^{\varphi}\bigl( \sigma\bigl( s_{1},X_{n}(s_{1})\bigr) - \sigma\bigl(s_{1},X_{j}(s_{1})\bigr)\bigr)\bigr\vert ^{2}\bigr]\,d{s_{1}} \\ \le& 3{a_{j + 1}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( { \mathbb {E} {{\bigl\vert {{X_{n}}(s_{1}) - {X_{j}}(s_{1})} \bigr\vert }^{2}}} \bigr)\,d{s_{1}}} \\ \le& 3{a_{j}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}( {3{a_{j}} + {C_{5}} {s_{1}}} )\,d{s_{1}}} \le{\psi_{1}}( t),\quad t \in[ {0,\eta} ]. \end{aligned}$$ </span></div></div><p> On the other hand, using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ16">3.14</a>) we arrive at </p><div id="Equap" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {{\tilde{\psi}}_{2,n}}( t) \le& \sup _{T_{2} \le s \le T_{2} + t} \mathbb{E} {\bigl\vert {{X_{n + 2}}( s) - {X_{j + 2}}( s)} \bigr\vert ^{2}} \\ \le& 3{a_{j + 2}} + {C_{6}} \int_{T_{2}}^{T_{2} + t} {\kappa\bigl( {\mathbb {E} {{\bigl\vert {{X_{n + 1}}(s_{1}) - {X_{j + 1}}(s_{1})} \bigr\vert }^{2}}} \bigr)\,d{s_{1}}} \\ \le& 3{a_{j + 2}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( {{{ \tilde{\psi}}_{1,n}}( t)} \bigr)\,d{s_{1}}} \\ \le& 3{a_{j + 1}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( {{ \psi_{1}}( t)} \bigr)\,d{s_{1}}} = {\psi_{2}}( t) \\ \le& 3{a_{j}}+{C_{5}}t = {\psi_{1}}( t),\quad t \in[ {0,\eta} ]. \end{aligned}$$ </span></div></div><p> Then we have proved </p><div id="Equaq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {\tilde{\psi}_{2,n}}( t) \le{\psi_{2}}( t) \le{ \psi_{1}}( t). \end{aligned}$$ </span></div></div> <p>Now assume that the assertion holds for <span class="mathjax-tex">$k \geq2$</span>. Then, by an analogous argument, one can obtain </p><div id="Equar" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} {{\tilde{\psi}}_{k + 1,n}}( t) \le& 3{a_{j + k + 1}} + \int _{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( { \mathbb{E} {{\bigl\vert {{X_{n + k}}(s_{1})-{X_{j + k}}(s_{1})} \bigr\vert }^{2}}} \bigr)\,d{s_{1}}} \\ \le& 3{a_{j + k + 1}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( {{{ \tilde{\psi}}_{k,n}}(s_{1})} \bigr)\,d{s_{1}}} \\ \le& 3{a_{j + k}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( {{\psi _{k}}(s_{1})} \bigr)\,d{s_{1}}} = { \psi_{k + 1}}( t) \\ \le& 3{a_{j + k - 1}} + \int_{T_{2}}^{T_{2} + t} {{\kappa_{2}}\bigl( {{\psi _{k - 1}}(s_{1})} \bigr)\,d{s_{1}}} \\ =& { \psi_{k}}( t),\quad t \in[ {0,\eta} ]. \end{aligned}$$ </span></div></div><p> Therefore, we obtain (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ17">3.15</a>) for all <i>k</i>. In terms of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ17">3.15</a>), we can define the function <span class="mathjax-tex">$\psi( t)$</span> by <span class="mathjax-tex">${\psi_{k}}( t) \downarrow\psi( t)$</span> (<span class="mathjax-tex">${k \to\infty}$</span>). We observe that </p><div id="Equas" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \psi( 0) =& \lim _{k \to\infty} {\psi_{k + 1}}( 0) \\ =& \lim _{k \to\infty} {a_{j + k}} = 0. \end{aligned}$$ </span></div></div><p> It is easy to verify that <span class="mathjax-tex">$\psi( t)$</span> is a continuous function on <span class="mathjax-tex">$[ {0,\eta} ]$</span>. Now by the definition of <span class="mathjax-tex">${\psi_{k + 1}}( t)$</span> and <span class="mathjax-tex">$\psi( t)$</span>, we have </p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \psi( t) =& \lim _{k \to\infty} {\psi_{k + 1}}( t) \\ =& \lim _{k \to\infty} \biggl[ {3{a_{j + k}} + \int_{0}^{t} {{\kappa_{2}}\bigl( {{ \psi_{k}}( s)} \bigr)\,ds} } \biggr] \\ =& \int_{0}^{t} {{\kappa_{2}}\bigl( {\psi( s)} \bigr)\,ds} . \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.16) </div></div><p> Since <span class="mathjax-tex">$\psi( 0) = 0$</span> and </p><div id="Equat" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \int_{0 + } {\frac{{dq}}{{{\kappa_{2}}( q)}}} = \frac {1}{{{C_{6}}}} \int_{0 + } {\frac{{dq}}{{\kappa( q)}}} = + \infty, \end{aligned}$$ </span></div></div><p> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ18">3.16</a>) implies <span class="mathjax-tex">$\psi( t) = 0$</span>, <span class="mathjax-tex">$t \in[ {0,\eta} ]$</span>.</p> <p>Therefore, we obtain </p><div id="Equau" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{k \to\infty} {{\tilde{\psi}}_{k,n}}( t) =& \lim _{k \to\infty} \sup _{0 \le s \le T_{2} + t } \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{j + k}}( s)}\bigr\vert ^{2}} \\ \le& \lim _{k \to\infty} \sup _{0 \le s \le T_{2}} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{j + k}}( s)}\bigr\vert ^{2}} \\ &{}+ \lim _{k \to\infty} \sup _{T_{2} \le s \le T_{2} + \eta} \mathbb{E} {\bigl\vert {{X_{n + k}}( s) - {X_{j +k}}( s)} \bigr\vert ^{2}} \\ \le& \lim _{k \to\infty} {\psi_{k}}( \eta) = \psi( \eta) = 0, \end{aligned}$$ </span></div></div><p> namely </p><div id="Equav" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le T_{2} + \eta} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div><p> But this conclusion is contradictory to the definition of <span class="mathjax-tex">$T_{2}$</span>. In other words, we have already shown that </p><div id="Equaw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \lim _{n,i \to\infty} \sup _{0 \le t \le T} \mathbb{E} {\bigl\vert {{X_{n}}( t) - {X_{i}}( t)} \bigr\vert ^{2}} = 0. \end{aligned}$$ </span></div></div><p> The proof of the existence of solutions of SDEs (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ1">1.1</a>) is complete. □</p> <h3 class="c-article__sub-heading" id="FPar13">Theorem 10</h3> <p> <i>Under the Hypothesis </i> <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a>, <i>the path</i>-<i>wise uniqueness holds for</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ1">1.1</a>), <span class="mathjax-tex">$t\in[0,T]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar14">Proof</h3> <p>Let <span class="mathjax-tex">$X( t)$</span> and <span class="mathjax-tex">$\tilde{X}( t)$</span> be two solutions of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ1">1.1</a>) on the same probability space and <span class="mathjax-tex">$X( 0) = \tilde{X}( 0)$</span>. We observe </p><div id="Equax" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \mathbb{E} {\bigl\vert {X( t) - \tilde{X}( t)} \bigr\vert ^{2}} \\& \quad = \mathbb{E} {\biggl\vert { \int_{0}^{t} {\bigl( {b\bigl( {s,X( s)} \bigr) - b \bigl( {s,\tilde{X}( s)} \bigr)} \bigr)\,ds} + \int_{0}^{t} {\bigl( {\sigma\bigl( {{s_{1}},X( s)} \bigr) - \sigma\bigl( {s,\tilde{X}( s)} \bigr)} \bigr)} \,d^{\circ}{B^{H}}( s)} \biggr\vert ^{2}} \\& \quad \le2\mathbb{E} {\biggl\vert { \int_{0}^{t} {\bigl({b\bigl( {s,X( s)} \bigr) - b \bigl( {s,\tilde{X}( s)} \bigr)} \bigr)\,ds} } \biggr\vert ^{2}} + 2 \mathbb{E} {\biggl\vert { \int_{0}^{t} {\bigl( {\sigma\bigl( {s,X( s)} \bigr) - \sigma\bigl( {s,\tilde{X}( s)} \bigr)} \bigr)} \,d^{\circ}{B^{H}}( s)} \biggr\vert ^{2}} \\& \quad \le8T\mathbb{E} \int_{0}^{t} \bigl(\bigl\vert {b\bigl( s, X( s) \bigr) - b\bigl( s,\tilde{X}( s) \bigr)\bigr\vert ^{2} + \bigl\vert \sigma\bigl( s,X( s) \bigr) - \sigma\bigl( s,\tilde{X}( s)\bigr)\bigr\vert }^{2} \\& \qquad {} + \bigl\vert D_{s}^{\varphi}\bigl( \sigma \bigl( s,X( s) \bigr) - \sigma\bigl( s,\tilde{X}( s) \bigr) \bigr) \bigr\vert ^{2}\bigr)\,ds. \end{aligned}$$ </span></div></div><p> Combining the above inequalities and the Hypothesis <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar4">4</a>, one has </p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {X( t) - \tilde{X}( t)} \bigr\vert ^{2}} \le8T \int_{0}^{t} {\kappa\bigl( {\mathbb{E} {{\bigl\vert {X( s) - \tilde{X}( s)} \bigr\vert }^{2}}} \bigr)}\,ds. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3.17) </div></div><p> Then, noticing that <span class="mathjax-tex">$\int_{0 + } {\frac{{dq}}{{\kappa( q)}}} = + \infty$</span>, the above inequality (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ19">3.17</a>) implies </p><div id="Equay" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {X( t) - \tilde{X}( t)} \bigr\vert ^{2}} = 0,\quad t\in[0,T]. \end{aligned}$$ </span></div></div> <p>Since <i>T</i> is an arbitrary positive number, we obtain from this <span class="mathjax-tex">$X( t) \equiv\tilde{X}( t)$</span>, for all <span class="mathjax-tex">$0\le t \le T$</span>.</p> <p>Thus the path-wise uniqueness holds for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ1">1.1</a>). □</p> </div></div></section> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1"> Øksendal, B: Stochastic Differential Equations. Springer, Berlin (2005) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1156.93406" aria-label="MATH reference 1">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20Differential%20Equations&publication_year=2005&author=%C3%98ksendal%2CB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2"> Arnold, L: Stochastic Differential Equations, Theory and Applications. John Wiley and Sons, New York (1974) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0278.60039" aria-label="MATH reference 2">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20Differential%20Equations%2C%20Theory%20and%20Applications&publication_year=1974&author=Arnold%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3"> Friedman, A: Stochastic Differential Equations and Applications. Dover Publications, New York (2006) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1113.60003" aria-label="MATH reference 3">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20Differential%20Equations%20and%20Applications&publication_year=2006&author=Friedman%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4"> Gard, T: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0628.60064" aria-label="MATH reference 4">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 4" href="http://scholar.google.com/scholar_lookup?&title=Introduction%20to%20Stochastic%20Differential%20Equations&publication_year=1988&author=Gard%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5"> Hurst, H: Long-term storage capacity in reservoirs. Trans. Am. Soc. Civ. Eng. <b>116</b>, 400-410 (1951) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=Long-term%20storage%20capacity%20in%20reservoirs&journal=Trans.%20Am.%20Soc.%20Civ.%20Eng.&volume=116&pages=400-410&publication_year=1951&author=Hurst%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6"> Kolmogorov, A: Wienersche spiralen und einige andere interessante kurven im Hilbertschen raum. C. R. (Dokl.) Acad. Sci. URSS <b>26</b>, 115-118 (1940) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3441" aria-label="MathSciNet reference 6">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?66.0552.03" aria-label="MATH reference 6">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=Wienersche%20spiralen%20und%20einige%20andere%20interessante%20kurven%20im%20Hilbertschen%20raum&journal=C.%20R.%20%28Dokl.%29%20Acad.%20Sci.%20URSS&volume=26&pages=115-118&publication_year=1940&author=Kolmogorov%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7"> Mandelbrot, B, Van Ness, J: Fractional Brownian motions, fractional noises and applications. SIAM Rev. <b>10</b>(4), 422-427 (1968) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/1010093" data-track-item_id="10.1137/1010093" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F1010093" aria-label="Article reference 7" data-doi="10.1137/1010093">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=242239" aria-label="MathSciNet reference 7">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0179.47801" aria-label="MATH reference 7">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Fractional%20Brownian%20motions%2C%20fractional%20noises%20and%20applications&journal=SIAM%20Rev.&doi=10.1137%2F1010093&volume=10&issue=4&pages=422-427&publication_year=1968&author=Mandelbrot%2CB&author=Ness%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8"> Chakravarti, N, Sebastian, K: Fractional Brownian motion models for polymers. Chem. Phys. Lett. <b>267</b>, 9-13 (1997) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/S0009-2614(97)00075-4" data-track-item_id="10.1016/S0009-2614(97)00075-4" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2FS0009-2614%2897%2900075-4" aria-label="Article reference 8" data-doi="10.1016/S0009-2614(97)00075-4">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Fractional%20Brownian%20motion%20models%20for%20polymers&journal=Chem.%20Phys.%20Lett.&doi=10.1016%2FS0009-2614%2897%2900075-4&volume=267&pages=9-13&publication_year=1997&author=Chakravarti%2CN&author=Sebastian%2CK"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9"> Hu, Y, Øksendal, B: Fractional white noise calculus and application to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. <b>6</b>, 1-32 (2003) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1142/S0219025703001110" data-track-item_id="10.1142/S0219025703001110" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1142%2FS0219025703001110" aria-label="Article reference 9" data-doi="10.1142/S0219025703001110">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1976868" aria-label="MathSciNet reference 9">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1045.60072" aria-label="MATH reference 9">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Fractional%20white%20noise%20calculus%20and%20application%20to%20finance&journal=Infin.%20Dimens.%20Anal.%20Quantum%20Probab.%20Relat.%20Top.&doi=10.1142%2FS0219025703001110&volume=6&pages=1-32&publication_year=2003&author=Hu%2CY&author=%C3%98ksendal%2CB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10"> Scheffer, R, Maciel, F: The fractional Brownian motion as a model for an industrial airlift reactor. Chem. Eng. Sci. <b>56</b>, 707-711 (2001) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/S0009-2509(00)00279-7" data-track-item_id="10.1016/S0009-2509(00)00279-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2FS0009-2509%2800%2900279-7" aria-label="Article reference 10" data-doi="10.1016/S0009-2509(00)00279-7">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=The%20fractional%20Brownian%20motion%20as%20a%20model%20for%20an%20industrial%20airlift%20reactor&journal=Chem.%20Eng.%20Sci.&doi=10.1016%2FS0009-2509%2800%2900279-7&volume=56&pages=707-711&publication_year=2001&author=Scheffer%2CR&author=Maciel%2CF"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11"> Lyons, T: Differential equations driven by rough signals. Rev. Mat. Iberoam. <b>14</b>, 215-310 (1998) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.4171/RMI/240" data-track-item_id="10.4171/RMI/240" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.4171%2FRMI%2F240" aria-label="Article reference 11" data-doi="10.4171/RMI/240">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1654527" aria-label="MathSciNet reference 11">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0923.34056" aria-label="MATH reference 11">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 11" href="http://scholar.google.com/scholar_lookup?&title=Differential%20equations%20driven%20by%20rough%20signals&journal=Rev.%20Mat.%20Iberoam.&doi=10.4171%2FRMI%2F240&volume=14&pages=215-310&publication_year=1998&author=Lyons%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12"> Nualart, D, Rascanu, A: Differential equations driven by fractional Brownian motion. Collect. Math. <b>53</b>, 55-81 (2002) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1893308" aria-label="MathSciNet reference 12">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1018.60057" aria-label="MATH reference 12">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=Differential%20equations%20driven%20by%20fractional%20Brownian%20motion&journal=Collect.%20Math.&volume=53&pages=55-81&publication_year=2002&author=Nualart%2CD&author=Rascanu%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13"> Mishura, Y: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/978-3-540-75873-0" data-track-item_id="10.1007/978-3-540-75873-0" data-track-value="book reference" data-track-action="book reference" href="https://link.springer.com/doi/10.1007/978-3-540-75873-0" aria-label="Book reference 13" data-doi="10.1007/978-3-540-75873-0">Book</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1138.60006" aria-label="MATH reference 13">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 13" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20Calculus%20for%20Fractional%20Brownian%20Motion%20and%20Related%20Processes&doi=10.1007%2F978-3-540-75873-0&publication_year=2008&author=Mishura%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14"> Biagini, F, Hu, Y, Oksendal, B, Zhang, T: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/978-1-84628-797-8" data-track-item_id="10.1007/978-1-84628-797-8" data-track-value="book reference" data-track-action="book reference" href="https://link.springer.com/doi/10.1007/978-1-84628-797-8" aria-label="Book reference 14" data-doi="10.1007/978-1-84628-797-8">Book</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1157.60002" aria-label="MATH reference 14">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 14" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20Calculus%20for%20Fractional%20Brownian%20Motion%20and%20Applications&doi=10.1007%2F978-1-84628-797-8&publication_year=2008&author=Biagini%2CF&author=Hu%2CY&author=Oksendal%2CB&author=Zhang%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15"> Cox, J, Ingersoll, J, Ross, S: A theory of the term structure of interest rate. Econometrica <b>53</b>, 385-407 (1985) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.2307/1911242" data-track-item_id="10.2307/1911242" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.2307%2F1911242" aria-label="Article reference 15" data-doi="10.2307/1911242">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=785475" aria-label="MathSciNet reference 15">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1274.91447" aria-label="MATH reference 15">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 15" href="http://scholar.google.com/scholar_lookup?&title=A%20theory%20of%20the%20term%20structure%20of%20interest%20rate&journal=Econometrica&doi=10.2307%2F1911242&volume=53&pages=385-407&publication_year=1985&author=Cox%2CJ&author=Ingersoll%2CJ&author=Ross%2CS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16"> Taniguchi, T: Successive approximations to solutions of stochastic differential equations. J. Differ. Equ. <b>96</b>, 152-169 (1992) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/0022-0396(92)90148-G" data-track-item_id="10.1016/0022-0396(92)90148-G" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2F0022-0396%2892%2990148-G" aria-label="Article reference 16" data-doi="10.1016/0022-0396(92)90148-G">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1153313" aria-label="MathSciNet reference 16">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0744.34052" aria-label="MATH reference 16">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=Successive%20approximations%20to%20solutions%20of%20stochastic%20differential%20equations&journal=J.%20Differ.%20Equ.&doi=10.1016%2F0022-0396%2892%2990148-G&volume=96&pages=152-169&publication_year=1992&author=Taniguchi%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17"> Kwok, Y: Pricing multi-asset options with an external barrier. Int. J. Theor. Appl. Finance <b>1</b>, 523-541 (1998) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1142/S021902499800028X" data-track-item_id="10.1142/S021902499800028X" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1142%2FS021902499800028X" aria-label="Article reference 17" data-doi="10.1142/S021902499800028X">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0987.91030" aria-label="MATH reference 17">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=Pricing%20multi-asset%20options%20with%20an%20external%20barrier&journal=Int.%20J.%20Theor.%20Appl.%20Finance&doi=10.1142%2FS021902499800028X&volume=1&pages=523-541&publication_year=1998&author=Kwok%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18"> Watanabe, S, Yamada, T: On the uniqueness of solution of stochastic differential equations II. J. Math. Kyoto Univ. <b>11</b>(3), 553-563 (1971) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=288876" aria-label="MathSciNet reference 18">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0229.60039" aria-label="MATH reference 18">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20uniqueness%20of%20solution%20of%20stochastic%20differential%20equations%20II&journal=J.%20Math.%20Kyoto%20Univ.&volume=11&issue=3&pages=553-563&publication_year=1971&author=Watanabe%2CS&author=Yamada%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19"> Barlow, M: One dimensional stochastic differential equations with no strong solution. J. Lond. Math. Soc. <b>26</b>, 335-347 (1982) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1112/jlms/s2-26.2.335" data-track-item_id="10.1112/jlms/s2-26.2.335" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1112%2Fjlms%2Fs2-26.2.335" aria-label="Article reference 19" data-doi="10.1112/jlms/s2-26.2.335">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=675177" aria-label="MathSciNet reference 19">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0456.60062" aria-label="MATH reference 19">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 19" href="http://scholar.google.com/scholar_lookup?&title=One%20dimensional%20stochastic%20differential%20equations%20with%20no%20strong%20solution&journal=J.%20Lond.%20Math.%20Soc.&doi=10.1112%2Fjlms%2Fs2-26.2.335&volume=26&pages=335-347&publication_year=1982&author=Barlow%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="20."><p class="c-article-references__text" id="ref-CR20"> Yamada, T: On a comparison theorem for solutions of stochastic differential equations and its applications. J. Math. Kyoto Univ. <b>13</b>(3), 497-512 (1973) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=339334" aria-label="MathSciNet reference 20">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0277.60047" aria-label="MATH reference 20">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 20" href="http://scholar.google.com/scholar_lookup?&title=On%20a%20comparison%20theorem%20for%20solutions%20of%20stochastic%20differential%20equations%20and%20its%20applications&journal=J.%20Math.%20Kyoto%20Univ.&volume=13&issue=3&pages=497-512&publication_year=1973&author=Yamada%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="21."><p class="c-article-references__text" id="ref-CR21"> Yamada, T: On the successive approximation of solutions of stochastic differential equations. J. Math. Kyoto Univ. <b>21</b>(3), 501-515 (1981) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=629781" aria-label="MathSciNet reference 21">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0484.60053" aria-label="MATH reference 21">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 21" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20successive%20approximation%20of%20solutions%20of%20stochastic%20differential%20equations&journal=J.%20Math.%20Kyoto%20Univ.&volume=21&issue=3&pages=501-515&publication_year=1981&author=Yamada%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="22."><p class="c-article-references__text" id="ref-CR22"> Yamada, T, Watanabe, S: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. <b>11</b>, 155-167 (1971) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=278420" aria-label="MathSciNet reference 22">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0236.60037" aria-label="MATH reference 22">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 22" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20uniqueness%20of%20solutions%20of%20stochastic%20differential%20equations&journal=J.%20Math.%20Kyoto%20Univ.&volume=11&pages=155-167&publication_year=1971&author=Yamada%2CT&author=Watanabe%2CS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="23."><p class="c-article-references__text" id="ref-CR23"> Carmona, P, Coutin, L, Montseny, G: Stochastic integration with respect to fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. <b>39</b>, 27-68 (2003) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/S0246-0203(02)01111-1" data-track-item_id="10.1016/S0246-0203(02)01111-1" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2FS0246-0203%2802%2901111-1" aria-label="Article reference 23" data-doi="10.1016/S0246-0203(02)01111-1">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1959841" aria-label="MathSciNet reference 23">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1016.60043" aria-label="MATH reference 23">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 23" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20integration%20with%20respect%20to%20fractional%20Brownian%20motion&journal=Ann.%20Inst.%20Henri%20Poincar%C3%A9%20Probab.%20Stat.&doi=10.1016%2FS0246-0203%2802%2901111-1&volume=39&pages=27-68&publication_year=2003&author=Carmona%2CP&author=Coutin%2CL&author=Montseny%2CG"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="24."><p class="c-article-references__text" id="ref-CR24"> Alòs, E, Mazet, O, Nualart, D: Stochastic calculus with respect to Gaussian process. Ann. Probab. <b>29</b>, 766-801 (2001) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1214/aop/1008956692" data-track-item_id="10.1214/aop/1008956692" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1214%2Faop%2F1008956692" aria-label="Article reference 24" data-doi="10.1214/aop/1008956692">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1849177" aria-label="MathSciNet reference 24">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1015.60047" aria-label="MATH reference 24">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 24" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20calculus%20with%20respect%20to%20Gaussian%20process&journal=Ann.%20Probab.&doi=10.1214%2Faop%2F1008956692&volume=29&pages=766-801&publication_year=2001&author=Al%C3%B2s%2CE&author=Mazet%2CO&author=Nualart%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="25."><p class="c-article-references__text" id="ref-CR25"> Duncan, T, Hu, Y, Pasik-Duncan, B: Stochastic calculus for fractional Brownian motion I: theory. SIAM J. Control Optim. <b>38</b>, 582-612 (2000) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/S036301299834171X" data-track-item_id="10.1137/S036301299834171X" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2FS036301299834171X" aria-label="Article reference 25" data-doi="10.1137/S036301299834171X">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1741154" aria-label="MathSciNet reference 25">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0947.60061" aria-label="MATH reference 25">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 25" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20calculus%20for%20fractional%20Brownian%20motion%20I%3A%20theory&journal=SIAM%20J.%20Control%20Optim.&doi=10.1137%2FS036301299834171X&volume=38&pages=582-612&publication_year=2000&author=Duncan%2CT&author=Hu%2CY&author=Pasik-Duncan%2CB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="26."><p class="c-article-references__text" id="ref-CR26"> Alòs, E, Nualart, D: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. <b>75</b>(3), 129-152 (2003) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1080/1045112031000078917" data-track-item_id="10.1080/1045112031000078917" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1080%2F1045112031000078917" aria-label="Article reference 26" data-doi="10.1080/1045112031000078917">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1978896" aria-label="MathSciNet reference 26">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1028.60048" aria-label="MATH reference 26">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 26" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20integration%20with%20respect%20to%20the%20fractional%20Brownian%20motion&journal=Stoch.%20Stoch.%20Rep.&doi=10.1080%2F1045112031000078917&volume=75&issue=3&pages=129-152&publication_year=2003&author=Al%C3%B2s%2CE&author=Nualart%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="27."><p class="c-article-references__text" id="ref-CR27"> Russo, F, Vallois, P: Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields <b>97</b>, 403-421 (1993) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF01195073" data-track-item_id="10.1007/BF01195073" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF01195073" aria-label="Article reference 27" data-doi="10.1007/BF01195073">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1245252" aria-label="MathSciNet reference 27">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0792.60046" aria-label="MATH reference 27">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 27" href="http://scholar.google.com/scholar_lookup?&title=Forward%2C%20backward%20and%20symmetric%20stochastic%20integration&journal=Probab.%20Theory%20Relat.%20Fields&doi=10.1007%2FBF01195073&volume=97&pages=403-421&publication_year=1993&author=Russo%2CF&author=Vallois%2CP"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="28."><p class="c-article-references__text" id="ref-CR28"> Xu, Y, Pei, B, Guo, R: Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst., Ser. B <b>20</b>, 2257-2267 (2015) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.3934/dcdsb.2015.20.2257" data-track-item_id="10.3934/dcdsb.2015.20.2257" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.3934%2Fdcdsb.2015.20.2257" aria-label="Article reference 28" data-doi="10.3934/dcdsb.2015.20.2257">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3423221" aria-label="MathSciNet reference 28">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1335.34090" aria-label="MATH reference 28">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 28" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20averaging%20for%20slow-fast%20dynamical%20systems%20with%20fractional%20Brownian%20motion&journal=Discrete%20Contin.%20Dyn.%20Syst.%2C%20Ser.%20B&doi=10.3934%2Fdcdsb.2015.20.2257&volume=20&pages=2257-2267&publication_year=2015&author=Xu%2CY&author=Pei%2CB&author=Guo%2CR"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="29."><p class="c-article-references__text" id="ref-CR29"> Xu, Y, Guo, R, et al.: Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst., Ser. B <b>19</b>(4), 1197-1212 (2014) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.3934/dcdsb.2014.19.1197" data-track-item_id="10.3934/dcdsb.2014.19.1197" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.3934%2Fdcdsb.2014.19.1197" aria-label="Article reference 29" data-doi="10.3934/dcdsb.2014.19.1197">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3206446" aria-label="MathSciNet reference 29">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1314.60122" aria-label="MATH reference 29">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 29" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20averaging%20principle%20for%20dynamical%20systems%20with%20fractional%20Brownian%20motion&journal=Discrete%20Contin.%20Dyn.%20Syst.%2C%20Ser.%20B&doi=10.3934%2Fdcdsb.2014.19.1197&volume=19&issue=4&pages=1197-1212&publication_year=2014&author=Xu%2CY&author=Guo%2CR"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="30."><p class="c-article-references__text" id="ref-CR30"> Xu, Y, Pei, B, Li, Y: An averaging principle for stochastic differential delay equations with fractional Brownian motion. Abstr. Appl. Anal. <b>2014</b>, Article ID 479195 (2014) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3166618" aria-label="MathSciNet reference 30">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 30" href="http://scholar.google.com/scholar_lookup?&title=An%20averaging%20principle%20for%20stochastic%20differential%20delay%20equations%20with%20fractional%20Brownian%20motion&journal=Abstr.%20Appl.%20Anal.&volume=2014&publication_year=2014&author=Xu%2CY&author=Pei%2CB&author=Li%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="31."><p class="c-article-references__text" id="ref-CR31"> Albeverio, S, Brzézniak, Z, Wu, J: Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. <b>371</b>, 309-322 (2010) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.jmaa.2010.05.039" data-track-item_id="10.1016/j.jmaa.2010.05.039" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.jmaa.2010.05.039" aria-label="Article reference 31" data-doi="10.1016/j.jmaa.2010.05.039">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2661009" aria-label="MathSciNet reference 31">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1197.60050" aria-label="MATH reference 31">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 31" href="http://scholar.google.com/scholar_lookup?&title=Existence%20of%20global%20solutions%20and%20invariant%20measures%20for%20stochastic%20differential%20equations%20driven%20by%20Poisson%20type%20noise%20with%20non-Lipschitz%20coefficients&journal=J.%20Math.%20Anal.%20Appl.&doi=10.1016%2Fj.jmaa.2010.05.039&volume=371&pages=309-322&publication_year=2010&author=Albeverio%2CS&author=Brz%C3%A9zniak%2CZ&author=Wu%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="32."><p class="c-article-references__text" id="ref-CR32"> Taniguchi, T: The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations. J. Math. Anal. Appl. <b>360</b>, 245-253 (2009) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.jmaa.2009.06.007" data-track-item_id="10.1016/j.jmaa.2009.06.007" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.jmaa.2009.06.007" aria-label="Article reference 32" data-doi="10.1016/j.jmaa.2009.06.007">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2548380" aria-label="MathSciNet reference 32">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1173.60021" aria-label="MATH reference 32">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 32" href="http://scholar.google.com/scholar_lookup?&title=The%20existence%20and%20uniqueness%20of%20energy%20solutions%20to%20local%20non-Lipschitz%20stochastic%20evolution%20equations&journal=J.%20Math.%20Anal.%20Appl.&doi=10.1016%2Fj.jmaa.2009.06.007&volume=360&pages=245-253&publication_year=2009&author=Taniguchi%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="33."><p class="c-article-references__text" id="ref-CR33"> Barbu, D, Bocsan, G: Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients. Czechoslov. Math. J. <b>52</b>, 87-95 (2002) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1023/A:1021723421437" data-track-item_id="10.1023/A:1021723421437" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1023%2FA%3A1021723421437" aria-label="Article reference 33" data-doi="10.1023/A:1021723421437">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1885459" aria-label="MathSciNet reference 33">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1001.60068" aria-label="MATH reference 33">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 33" href="http://scholar.google.com/scholar_lookup?&title=Approximations%20to%20mild%20solutions%20of%20stochastic%20semilinear%20equations%20with%20non-Lipschitz%20coefficients&journal=Czechoslov.%20Math.%20J.&doi=10.1023%2FA%3A1021723421437&volume=52&pages=87-95&publication_year=2002&author=Barbu%2CD&author=Bocsan%2CG"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="34."><p class="c-article-references__text" id="ref-CR34"> Xu, Y, Pei, B, Wu, J: Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. Stoch. Dyn. (2016). doi:<a href="https://doi.org/10.1142/S0219493717500137" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1142/S0219493717500137">10.1142/S0219493717500137</a> </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 34" href="http://scholar.google.com/scholar_lookup?&title=Stochastic%20averaging%20principle%20for%20differential%20equations%20with%20non-Lipschitz%20coefficients%20driven%20by%20fractional%20Brownian%20motion&journal=Stoch.%20Dyn.&doi=10.1142%2FS0219493717500137&publication_year=2016&author=Xu%2CY&author=Pei%2CB&author=Wu%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="35."><p class="c-article-references__text" id="ref-CR35"> Xu, Y, Pei, B, Guo, G: Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise. Appl. Math. Comput. <b>263</b>, 398-409 (2015) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3348553" aria-label="MathSciNet reference 35">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 35" href="http://scholar.google.com/scholar_lookup?&title=Existence%20and%20stability%20of%20solutions%20to%20non-Lipschitz%20stochastic%20differential%20equations%20driven%20by%20L%C3%A9vy%20noise&journal=Appl.%20Math.%20Comput.&volume=263&pages=398-409&publication_year=2015&author=Xu%2CY&author=Pei%2CB&author=Guo%2CG"> Google Scholar</a> </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1186/s13662-016-0916-1?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>This work was supported by the NSF of China (11572247), the Fundamental Research Funds for the Central Universities and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University. Authors would like to thank the referees for their helpful comments.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China</p><p class="c-article-author-affiliation__authors-list">Bin Pei & Yong Xu</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Bin-Pei-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Bin Pei</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="https://www.biomedcentral.com/search?query=author%23Bin%20Pei" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Bin%20Pei" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Bin%20Pei%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Yong-Xu-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Yong Xu</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="https://www.biomedcentral.com/search?query=author%23Yong%20Xu" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Yong%20Xu" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Yong%20Xu%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:hsux3@nwpu.edu.cn">Yong Xu</a>.</p></div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><h3 class="c-article__sub-heading">Competing interests</h3><p>The authors declare that they have no competing interests.</p><h3 class="c-article__sub-heading">Authors’ contributions</h3><p>All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.</p></div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendix</h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading u-visually-hidden" id="App1">Appendix</h3> <h3 class="c-article__sub-heading" id="FPar15">Proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar5">5</a> </h3> <p>Since <span class="mathjax-tex">$\kappa( q)$</span> is a concave and non-negative function, we can choose two positive constants <span class="mathjax-tex">$a > 0$</span> and <span class="mathjax-tex">$b > 0$</span>, so that </p><div id="Equaz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\kappa(q) \le a + bq,\quad q \ge0, $$</span></div></div><p> then, by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ2">2.1</a>), we get </p><div id="Equba" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \mathbb{E} {\bigl\vert {\sigma( {t,X})} \bigr\vert ^{2}} + \mathbb{E} {\bigl\vert {b( {t,X})}\bigr\vert ^{2}} + \mathbb{E} { \bigl\vert {D_{t}^{\varphi}\sigma( {t,X})} \bigr\vert ^{2}} \\& \quad \le2\mathbb{E}\bigl( {{{\bigl\vert {\sigma( {t,0})} \bigr\vert }^{2}} + {{\bigl\vert {b( {t,0})}\bigr\vert }^{2}}} +{ \bigl\vert {D_{t}^{\varphi}\sigma( {t,0})} \bigr\vert ^{2}}\bigr)+ 2\mathbb{E} {\bigl\vert {\sigma( {t,X}) - \sigma( {t,0})} \bigr\vert ^{2}} \\& \qquad {} + 2\mathbb{E} {\bigl\vert {b( {t,X}) - b( {t,0})} \bigr\vert ^{2}} + 2\mathbb{E} {\bigl\vert {D_{t}^{\varphi}(\sigma( {t,X})-\sigma({t,0}))} \bigr\vert ^{2}} \\& \quad \le2\sup _{0 \le t \le T} \mathbb{E}\bigl( {{{\bigl\vert {\sigma( {t,0})} \bigr\vert }^{2}} + {{\bigl\vert {b( {t,0})} \bigr\vert }^{2}} + {{\bigl\vert {D_{t}^{\varphi}\sigma( {t,0})} \bigr\vert }^{2}}} \bigr) + 2\kappa\bigl( {\mathbb{E} {{\vert X \vert }^{2}}} \bigr) \\& \quad \le K\bigl( {1 + \mathbb{E} {{\vert X \vert }^{2}}} \bigr), \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$K = \max[ {2\sup _{0 \le t \le T} \mathbb {E}( {{{\vert {\sigma( {t,0})} \vert }^{2}} + {{\vert {b( {t,0})} \vert }^{2}} + {{\vert {D_{t}^{\varphi}\sigma( {t,0})} \vert }^{2}}}) + 2a,2b} ] < + \infty$</span>. □</p> <h3 class="c-article__sub-heading" id="FPar16">Proof of Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar6">6</a> </h3> <p>Using mathematical induction, we first assume that </p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{k}}( t)} \bigr\vert ^{2}} \le3\mathbb{E} {\vert \xi \vert ^{2}}\sum _{l = 0}^{k} {\frac{{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}} + \sum_{l = 1}^{k} {\frac{{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}} \end{aligned}$$ </span></div><div class="c-article-equation__number"> (A.1) </div></div><p> holds, <span class="mathjax-tex">$t \in[0,T]$</span>, <span class="mathjax-tex">$k = 1,2, \ldots$</span> .</p> <p>Clearly, by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar3">3</a> and Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-016-0916-1#FPar5">5</a>, we arrive at </p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{1}}( t)} \bigr\vert ^{2}} \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 3 \mathbb{E} {\biggl\vert { \int_{0}^{t} {b\bigl( {s,{X_{0}}( s)} \bigr)}\,ds} \biggr\vert ^{2}} + 3\mathbb{E} {\biggl\vert { \int_{0}^{t} {\sigma\bigl( {s,{X_{0}}( s)} \bigr)} \,d^{\circ}{B^{H}}( s)} \biggr\vert ^{2}} \\ \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 12T\mathbb{E} \int_{0}^{t} {\bigl( {{{ \bigl\vert {b \bigl({s,{X_{0}}( s)} \bigr)} \bigr\vert }^{2}} + {{\bigl\vert {\sigma\bigl( {s,{X_{0}}( s)} \bigr)} \bigr\vert }^{2}} + {{\bigl\vert {D_{s}^{\varphi}\sigma\bigl( {s,{X_{0}}( s)} \bigr)} \bigr\vert }^{2}}} \bigr)\,ds} \\ \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 12KTt\bigl( {1 + \mathbb{E} {{\vert \xi \vert }^{2}}} \bigr). \end{aligned}$$ </span></div><div class="c-article-equation__number"> (A.2) </div></div><p> Now, assume that (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ20">A.1</a>) holds for <i>k</i>, then we have, for <span class="mathjax-tex">$k+1$</span>, </p><div id="Equbb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathbb{E} {\bigl\vert {{X_{k + 1}}( t)} \bigr\vert ^{2}} \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 3\mathbb{E} {\biggl\vert { \int_{0}^{t} {b\bigl( {s,{X_{k}}( s)} \bigr)}\,ds} \biggr\vert ^{2}} + 3\mathbb{E} {\biggl\vert { \int_{0}^{t} {\sigma\bigl( {s,{X_{k}}( s)} \bigr)} \,d^{\circ}{B^{H}}( s)} \biggr\vert ^{2}} \\ \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 12T\mathbb{E} \int_{0}^{t} {\bigl( {{{ \bigl\vert {b \bigl({s,{X_{k}}( s)} \bigr)} \bigr\vert }^{2}} + {{\bigl\vert {\sigma\bigl( {s,{X_{k}}( s)} \bigr)} \bigr\vert }^{2}} + {{\bigl\vert {D_{s}^{\varphi}\sigma\bigl( {s,{X_{k}}( s)} \bigr)} \bigr\vert }^{2}}} \bigr)}\,ds \\ \le& 3\mathbb{E} {\vert \xi \vert ^{2}} + 12KT \int_{0}^{t} {\bigl( {1 + \mathbb{E} {{\bigl\vert {{X_{k}}( s)} \bigr\vert }^{2}}} \bigr)}\,ds \\ \le& 3 \mathbb{E} {\vert \xi \vert ^{2}} + 12KT \int_{0}^{t} {\Biggl( {1 + 3\mathbb{E} {{\vert \xi \vert }^{2}}\sum_{l = 0}^{k} { \frac{{{{( {12KT})}^{l}}}}{{l!}}} {s^{l}} + \sum_{l = 1}^{k} {\frac{{{{( {12KT})}^{l}}}}{{l!}}} {s^{l}}} \Biggr)}\,ds \\ =& 3\mathbb{E} {\vert \xi \vert ^{2}} + 12KTt + 3\mathbb{E} {\vert \xi \vert ^{2}}\sum_{l = 1}^{k + 1} { \frac{{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}} + \sum_{l = 2}^{k + 1} {\frac{{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}} \\ =& 3\mathbb{E} {\vert \xi \vert ^{2}}\sum_{l = 0}^{k + 1} { \frac{{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}} + \sum_{l = 1}^{k + 1} {\frac {{{{( {12KT})}^{l}}}}{{l!}}} {t^{l}}. \end{aligned}$$ </span></div></div><p> Therefore, by induction, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ20">A.1</a>) holds for all <i>k</i>.</p> <p>Now, we obtain the form <span class="mathjax-tex">${C_{1}} = 3( {1 + \mathbb{E}{{\vert \xi \vert }^{2}}} )\exp( {12K{T^{2}}})$</span>, then (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ21">A.2</a>) implies (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-016-0916-1#Equ4">3.2</a>). □</p> </div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p> <b>Open Access</b> This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.</p> <p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=On%20the%20non-Lipschitz%20stochastic%20differential%20equations%20driven%20by%20fractional%20Brownian%20motion&author=Bin%20Pei%20et%20al&contentID=10.1186%2Fs13662-016-0916-1&copyright=Pei%20and%20Xu&publication=1687-1847&publicationDate=2016-07-23&publisherName=SpringerNature&orderBeanReset=true&oa=CC%20BY">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1186/s13662-016-0916-1" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1186/s13662-016-0916-1" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. Verify currency and authenticity via CrossMark" src="data:image/svg+xml;base64,<svg height="81" width="57" xmlns="http://www.w3.org/2000/svg"><g fill="none" fill-rule="evenodd"><path d="m17.35 35.45 21.3-14.2v-17.03h-21.3" fill="#989898"/><path d="m38.65 35.45-21.3-14.2v-17.03h21.3" fill="#747474"/><path d="m28 .5c-12.98 0-23.5 10.52-23.5 23.5s10.52 23.5 23.5 23.5 23.5-10.52 23.5-23.5c0-6.23-2.48-12.21-6.88-16.62-4.41-4.4-10.39-6.88-16.62-6.88zm0 41.25c-9.8 0-17.75-7.95-17.75-17.75s7.95-17.75 17.75-17.75 17.75 7.95 17.75 17.75c0 4.71-1.87 9.22-5.2 12.55s-7.84 5.2-12.55 5.2z" fill="#535353"/><path d="m41 36c-5.81 6.23-15.23 7.45-22.43 2.9-7.21-4.55-10.16-13.57-7.03-21.5l-4.92-3.11c-4.95 10.7-1.19 23.42 8.78 29.71 9.97 6.3 23.07 4.22 30.6-4.86z" fill="#9c9c9c"/><path d="m.2 58.45c0-.75.11-1.42.33-2.01s.52-1.09.91-1.5c.38-.41.83-.73 1.34-.94.51-.22 1.06-.32 1.65-.32.56 0 1.06.11 1.51.35.44.23.81.5 1.1.81l-.91 1.01c-.24-.24-.49-.42-.75-.56-.27-.13-.58-.2-.93-.2-.39 0-.73.08-1.05.23-.31.16-.58.37-.81.66-.23.28-.41.63-.53 1.04-.13.41-.19.88-.19 1.39 0 1.04.23 1.86.68 2.46.45.59 1.06.88 1.84.88.41 0 .77-.07 1.07-.23s.59-.39.85-.68l.91 1c-.38.43-.8.76-1.28.99-.47.22-1 .34-1.58.34-.59 0-1.13-.1-1.64-.31-.5-.2-.94-.51-1.31-.91-.38-.4-.67-.9-.88-1.48-.22-.59-.33-1.26-.33-2.02zm8.4-5.33h1.61v2.54l-.05 1.33c.29-.27.61-.51.96-.72s.76-.31 1.24-.31c.73 0 1.27.23 1.61.71.33.47.5 1.14.5 2.02v4.31h-1.61v-4.1c0-.57-.08-.97-.25-1.21-.17-.23-.45-.35-.83-.35-.3 0-.56.08-.79.22-.23.15-.49.36-.78.64v4.8h-1.61zm7.37 6.45c0-.56.09-1.06.26-1.51.18-.45.42-.83.71-1.14.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.36c.07.62.29 1.1.65 1.44.36.33.82.5 1.38.5.29 0 .57-.04.83-.13s.51-.21.76-.37l.55 1.01c-.33.21-.69.39-1.09.53-.41.14-.83.21-1.26.21-.48 0-.92-.08-1.34-.25-.41-.16-.76-.4-1.07-.7-.31-.31-.55-.69-.72-1.13-.18-.44-.26-.95-.26-1.52zm4.6-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.07.45-.31.29-.5.73-.58 1.3zm2.5.62c0-.57.09-1.08.28-1.53.18-.44.43-.82.75-1.13s.69-.54 1.1-.71c.42-.16.85-.24 1.31-.24.45 0 .84.08 1.17.23s.61.34.85.57l-.77 1.02c-.19-.16-.38-.28-.56-.37-.19-.09-.39-.14-.61-.14-.56 0-1.01.21-1.35.63-.35.41-.52.97-.52 1.67 0 .69.17 1.24.51 1.66.34.41.78.62 1.32.62.28 0 .54-.06.78-.17.24-.12.45-.26.64-.42l.67 1.03c-.33.29-.69.51-1.08.65-.39.15-.78.23-1.18.23-.46 0-.9-.08-1.31-.24-.4-.16-.75-.39-1.05-.7s-.53-.69-.7-1.13c-.17-.45-.25-.96-.25-1.53zm6.91-6.45h1.58v6.17h.05l2.54-3.16h1.77l-2.35 2.8 2.59 4.07h-1.75l-1.77-2.98-1.08 1.23v1.75h-1.58zm13.69 1.27c-.25-.11-.5-.17-.75-.17-.58 0-.87.39-.87 1.16v.75h1.34v1.27h-1.34v5.6h-1.61v-5.6h-.92v-1.2l.92-.07v-.72c0-.35.04-.68.13-.98.08-.31.21-.57.4-.79s.42-.39.71-.51c.28-.12.63-.18 1.04-.18.24 0 .48.02.69.07.22.05.41.1.57.17zm.48 5.18c0-.57.09-1.08.27-1.53.17-.44.41-.82.72-1.13.3-.31.65-.54 1.04-.71.39-.16.8-.24 1.23-.24s.84.08 1.24.24c.4.17.74.4 1.04.71s.54.69.72 1.13c.19.45.28.96.28 1.53s-.09 1.08-.28 1.53c-.18.44-.42.82-.72 1.13s-.64.54-1.04.7-.81.24-1.24.24-.84-.08-1.23-.24-.74-.39-1.04-.7c-.31-.31-.55-.69-.72-1.13-.18-.45-.27-.96-.27-1.53zm1.65 0c0 .69.14 1.24.43 1.66.28.41.68.62 1.18.62.51 0 .9-.21 1.19-.62.29-.42.44-.97.44-1.66 0-.7-.15-1.26-.44-1.67-.29-.42-.68-.63-1.19-.63-.5 0-.9.21-1.18.63-.29.41-.43.97-.43 1.67zm6.48-3.44h1.33l.12 1.21h.05c.24-.44.54-.79.88-1.02.35-.24.7-.36 1.07-.36.32 0 .59.05.78.14l-.28 1.4-.33-.09c-.11-.01-.23-.02-.38-.02-.27 0-.56.1-.86.31s-.55.58-.77 1.1v4.2h-1.61zm-47.87 15h1.61v4.1c0 .57.08.97.25 1.2.17.24.44.35.81.35.3 0 .57-.07.8-.22.22-.15.47-.39.73-.73v-4.7h1.61v6.87h-1.32l-.12-1.01h-.04c-.3.36-.63.64-.98.86-.35.21-.76.32-1.24.32-.73 0-1.27-.24-1.61-.71-.33-.47-.5-1.14-.5-2.02zm9.46 7.43v2.16h-1.61v-9.59h1.33l.12.72h.05c.29-.24.61-.45.97-.63.35-.17.72-.26 1.1-.26.43 0 .81.08 1.15.24.33.17.61.4.84.71.24.31.41.68.53 1.11.13.42.19.91.19 1.44 0 .59-.09 1.11-.25 1.57-.16.47-.38.85-.65 1.16-.27.32-.58.56-.94.73-.35.16-.72.25-1.1.25-.3 0-.6-.07-.9-.2s-.59-.31-.87-.56zm0-2.3c.26.22.5.37.73.45.24.09.46.13.66.13.46 0 .84-.2 1.15-.6.31-.39.46-.98.46-1.77 0-.69-.12-1.22-.35-1.61-.23-.38-.61-.57-1.13-.57-.49 0-.99.26-1.52.77zm5.87-1.69c0-.56.08-1.06.25-1.51.16-.45.37-.83.65-1.14.27-.3.58-.54.93-.71s.71-.25 1.08-.25c.39 0 .73.07 1 .2.27.14.54.32.81.55l-.06-1.1v-2.49h1.61v9.88h-1.33l-.11-.74h-.06c-.25.25-.54.46-.88.64-.33.18-.69.27-1.06.27-.87 0-1.56-.32-2.07-.95s-.76-1.51-.76-2.65zm1.67-.01c0 .74.13 1.31.4 1.7.26.38.65.58 1.15.58.51 0 .99-.26 1.44-.77v-3.21c-.24-.21-.48-.36-.7-.45-.23-.08-.46-.12-.7-.12-.45 0-.82.19-1.13.59-.31.39-.46.95-.46 1.68zm6.35 1.59c0-.73.32-1.3.97-1.71.64-.4 1.67-.68 3.08-.84 0-.17-.02-.34-.07-.51-.05-.16-.12-.3-.22-.43s-.22-.22-.38-.3c-.15-.06-.34-.1-.58-.1-.34 0-.68.07-1 .2s-.63.29-.93.47l-.59-1.08c.39-.24.81-.45 1.28-.63.47-.17.99-.26 1.54-.26.86 0 1.51.25 1.93.76s.63 1.25.63 2.21v4.07h-1.32l-.12-.76h-.05c-.3.27-.63.48-.98.66s-.73.27-1.14.27c-.61 0-1.1-.19-1.48-.56-.38-.36-.57-.85-.57-1.46zm1.57-.12c0 .3.09.53.27.67.19.14.42.21.71.21.28 0 .54-.07.77-.2s.48-.31.73-.56v-1.54c-.47.06-.86.13-1.18.23-.31.09-.57.19-.76.31s-.33.25-.41.4c-.09.15-.13.31-.13.48zm6.29-3.63h-.98v-1.2l1.06-.07.2-1.88h1.34v1.88h1.75v1.27h-1.75v3.28c0 .8.32 1.2.97 1.2.12 0 .24-.01.37-.04.12-.03.24-.07.34-.11l.28 1.19c-.19.06-.4.12-.64.17-.23.05-.49.08-.76.08-.4 0-.74-.06-1.02-.18-.27-.13-.49-.3-.67-.52-.17-.21-.3-.48-.37-.78-.08-.3-.12-.64-.12-1.01zm4.36 2.17c0-.56.09-1.06.27-1.51s.41-.83.71-1.14c.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.37c.08.62.29 1.1.65 1.44.36.33.82.5 1.38.5.3 0 .58-.04.84-.13.25-.09.51-.21.76-.37l.54 1.01c-.32.21-.69.39-1.09.53s-.82.21-1.26.21c-.47 0-.92-.08-1.33-.25-.41-.16-.77-.4-1.08-.7-.3-.31-.54-.69-.72-1.13-.17-.44-.26-.95-.26-1.52zm4.61-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.08.45-.31.29-.5.73-.57 1.3zm3.01 2.23c.31.24.61.43.92.57.3.13.63.2.98.2.38 0 .65-.08.83-.23s.27-.35.27-.6c0-.14-.05-.26-.13-.37-.08-.1-.2-.2-.34-.28-.14-.09-.29-.16-.47-.23l-.53-.22c-.23-.09-.46-.18-.69-.3-.23-.11-.44-.24-.62-.4s-.33-.35-.45-.55c-.12-.21-.18-.46-.18-.75 0-.61.23-1.1.68-1.49.44-.38 1.06-.57 1.83-.57.48 0 .91.08 1.29.25s.71.36.99.57l-.74.98c-.24-.17-.49-.32-.73-.42-.25-.11-.51-.16-.78-.16-.35 0-.6.07-.76.21-.17.15-.25.33-.25.54 0 .14.04.26.12.36s.18.18.31.26c.14.07.29.14.46.21l.54.19c.23.09.47.18.7.29s.44.24.64.4c.19.16.34.35.46.58.11.23.17.5.17.82 0 .3-.06.58-.17.83-.12.26-.29.48-.51.68-.23.19-.51.34-.84.45-.34.11-.72.17-1.15.17-.48 0-.95-.09-1.41-.27-.46-.19-.86-.41-1.2-.68z" fill="#535353"/></g></svg>"></a></div><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Pei, B., Xu, Y. On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion. <i>Adv Differ Equ</i> <b>2016</b>, 194 (2016). https://doi.org/10.1186/s13662-016-0916-1</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1186/s13662-016-0916-1?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-02-09">09 February 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-06-30">30 June 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-07-23">23 July 2016</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1186/s13662-016-0916-1</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" hidden=""><h3 class="c-article__sub-heading">Share this article</h3><p class="c-article-share-box__description">Anyone you share the following link with will be able to read this content:</p><button class="js-get-share-url c-article-share-box__button" type="button" id="get-share-url" data-track="click" data-track-label="button" data-track-external="" data-track-action="get shareable link">Get shareable link</button><div class="js-no-share-url-container u-display-none" hidden=""><p class="js-c-article-share-box__no-sharelink-info c-article-share-box__no-sharelink-info">Sorry, a shareable link is not currently available for this article.</p></div><div class="js-share-url-container u-display-none" hidden=""><p class="js-share-url c-article-share-box__only-read-input" id="share-url" data-track="click" data-track-label="button" data-track-action="select share url"></p><button class="js-copy-share-url c-article-share-box__button--link-like" type="button" id="copy-share-url" data-track="click" data-track-label="button" data-track-action="copy share url" data-track-external="">Copy to clipboard</button></div><p class="js-c-article-share-box__additional-info c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=fractional%20Brownian%20motion&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">fractional Brownian motion</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=existence%20and%20uniqueness&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">existence and uniqueness</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=stochastic%20differential%20equations&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">stochastic differential equations</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=non-Lipschitz%20condition&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">non-Lipschitz condition</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </article> </main> <div class="c-article-extras u-text-sm u-hide-print" data-container-type="reading-companion" data-track-component="reading companion"> <aside> <div data-test="download-article-link-wrapper" class="js-context-bar-sticky-point-desktop" data-track-context="reading companion"> <div class="c-pdf-download u-clear-both"> <a href="//advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-016-0916-1.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="link" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-download"/></svg> </a> </div> </div> <div class="c-reading-companion"> <div class="c-reading-companion__sticky" data-component="reading-companion-sticky" data-test="reading-companion-sticky"> <div class="c-reading-companion__panel c-reading-companion__sections c-reading-companion__panel--active" id="tabpanel-sections"> <div class="js-ad u-lazy-ad-wrapper u-mt-16 u-hide" data-component-mpu> <aside class="adsbox c-ad c-ad--300x250 u-mt-16" data-component-mpu> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-MPU1" data-ad-type="MPU1" data-test="MPU1-ad" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springer_open/advancesincontinuousanddiscretemodels/articles" data-gpt-sizes="300x250" data-gpt-targeting="pos=MPU1;doi=10.1186/s13662-016-0916-1;type=article;kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition;pmc=M12031,M00009,M12007,M12066,M12147,M12155;" > <noscript> <a href="//pubads.g.doubleclick.net/gampad/jump?iu=/270604982/springer_open/advancesincontinuousanddiscretemodels/articles&sz=300x250&pos=MPU1&doi=10.1186/s13662-016-0916-1&type=article&kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition&pmc=M12031,M00009,M12007,M12066,M12147,M12155&"> <img data-test="gpt-advert-fallback-img" src="//pubads.g.doubleclick.net/gampad/ad?iu=/270604982/springer_open/advancesincontinuousanddiscretemodels/articles&sz=300x250&pos=MPU1&doi=10.1186/s13662-016-0916-1&type=article&kwrd=fractional Brownian motion,existence and uniqueness,stochastic differential equations,non-Lipschitz condition&pmc=M12031,M00009,M12007,M12066,M12147,M12155&" alt="Advertisement" width="300" height="250"> </a> </noscript> </div> </div> </aside> </div> </div> <div class="c-reading-companion__panel c-reading-companion__figures c-reading-companion__panel--full-width" id="tabpanel-figures"></div> <div class="c-reading-companion__panel c-reading-companion__references c-reading-companion__panel--full-width" id="tabpanel-references"></div> </div> </div> </aside> </div> </div> <img rel="nofollow" class='tracker' style='display:none' src='/track/article/10.1186/s13662-016-0916-1' alt=""/> <footer> <div class="c-publisher-footer u-color-inherit" data-test="publisher-footer"> <div class="u-container"> <div class="u-display-flex u-flex-wrap u-justify-content-space-between" data-test="publisher-footer-menu"> <div class="u-display-flex"> <ul class="c-list-group c-list-group--sm u-mr-24 u-mb-16"> <li class="c-list-group__item"> <a class="u-gray-link" href="https://support.biomedcentral.com/support/home">Support and Contact</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="//www.springeropen.com/about/jobs">Jobs</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="https://authorservices.springernature.com/language-editing/">Language editing for authors</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="https://authorservices.springernature.com/scientific-editing/">Scientific editing for authors</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="https://biomedcentral.typeform.com/to/VLXboo">Leave feedback</a> </li> </ul> <ul class="c-list-group c-list-group--sm u-mr-24 u-mb-16"> <li class="c-list-group__item"> <a class="u-gray-link" href="//www.springeropen.com/terms-and-conditions">Terms and conditions</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="//www.springeropen.com/privacy-statement">Privacy statement</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="//www.springeropen.com/accessibility">Accessibility</a> </li> <li class="c-list-group__item"> <a class="u-gray-link" href="//www.springeropen.com/cookies">Cookies</a> </li> </ul> </div> <div class="u-mb-24"> <h3 id="social-menu" class="u-text-sm u-reset-margin u-text-normal">Follow SpringerOpen</h3> <ul class="u-display-flex u-list-reset" data-test="footer-social-links"> <li class="u-mt-8 u-mr-8"> <a href="https://twitter.com/springeropen" data-track="click" data-track-category="Social" data-track-action="Clicked SpringerOpen Twitter" class="u-gray-link"> <span class="u-visually-hidden">SpringerOpen Twitter page</span> <svg class="u-icon u-text-lg" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-twitter-bordered"></use> </svg> </a> </li> <li class="u-mt-8 u-mr-8"> <a href="https://www.facebook.com/SpringerOpn" data-track="click" data-track-category="Social" data-track-action="Clicked SpringerOpen Facebook" class="u-gray-link"> <span class="u-visually-hidden">SpringerOpen Facebook page</span> <svg class="u-icon u-text-lg" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-facebook-bordered"></use> </svg> </a> </li> </ul> </div> </div> <p class="u-reset-margin"> By using this website, you agree to our <a class="u-gray-link" href="//www.springeropen.com/terms-and-conditions">Terms and Conditions</a>, <a class="u-gray-link" href="https://www.springernature.com/ccpa">Your US state privacy rights</a>, <a class="u-gray-link" href="//www.springeropen.com/privacy-statement">Privacy statement</a> and <a class="u-gray-link" href="//www.springeropen.com/cookies" data-test="cookie-link">Cookies</a> policy. <a class="u-gray-link" data-cc-action="preferences" href="javascript:void(0);">Your privacy choices/Manage cookies</a> we use in the preference centre. </p> </div> </div> <div class="c-corporate-footer"> <div class="u-container"> <img src=/static/images/logo-springernature-acb40b85fb.svg class="c-corporate-footer__logo" alt="Springer Nature" itemprop="logo" role="img"> <p class="c-corporate-footer__legal" data-test="copyright"> © 2025 BioMed Central Ltd unless otherwise stated. Part of <a class="c-corporate-footer__link" href="https://www.springernature.com" itemscope itemtype="http://schema.org/Organization" itemid="#parentOrganization">Springer Nature</a>. </p> </div> </div> </footer> </div> <div class="u-visually-hidden" aria-hidden="true"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="a" d="M0 .74h56.72v55.24H0z"/></defs><symbol id="icon-access" viewBox="0 0 18 18"><path d="m14 8c.5522847 0 1 .44771525 1 1v7h2.5c.2761424 0 .5.2238576.5.5v1.5h-18v-1.5c0-.2761424.22385763-.5.5-.5h2.5v-7c0-.55228475.44771525-1 1-1s1 .44771525 1 1v6.9996556h8v-6.9996556c0-.55228475.4477153-1 1-1zm-8 0 2 1v5l-2 1zm6 0v7l-2-1v-5zm-2.42653766-7.59857636 7.03554716 4.92488299c.4162533.29137735.5174853.86502537.226108 1.28127873-.1721584.24594054-.4534847.39241464-.7536934.39241464h-14.16284822c-.50810197 0-.92-.41189803-.92-.92 0-.30020869.1464741-.58153499.39241464-.75369337l7.03554714-4.92488299c.34432015-.2410241.80260453-.2410241 1.14692468 0zm-.57346234 2.03988748-3.65526982 2.55868888h7.31053962z" fill-rule="evenodd"/></symbol><symbol id="icon-account" viewBox="0 0 18 18"><path d="m10.2379028 16.9048051c1.3083556-.2032362 2.5118471-.7235183 3.5294683-1.4798399-.8731327-2.5141501-2.0638925-3.935978-3.7673711-4.3188248v-1.27684611c1.1651924-.41183641 2-1.52307546 2-2.82929429 0-1.65685425-1.3431458-3-3-3-1.65685425 0-3 1.34314575-3 3 0 1.30621883.83480763 2.41745788 2 2.82929429v1.27684611c-1.70347856.3828468-2.89423845 1.8046747-3.76737114 4.3188248 1.01762123.7563216 2.22111275 1.2766037 3.52946833 1.4798399.40563808.0629726.81921174.0951949 1.23790281.0951949s.83226473-.0322223 1.2379028-.0951949zm4.3421782-2.1721994c1.4927655-1.4532925 2.419919-3.484675 2.419919-5.7326057 0-4.418278-3.581722-8-8-8s-8 3.581722-8 8c0 2.2479307.92715352 4.2793132 2.41991895 5.7326057.75688473-2.0164459 1.83949951-3.6071894 3.48926591-4.3218837-1.14534283-.70360829-1.90918486-1.96796271-1.90918486-3.410722 0-2.209139 1.790861-4 4-4s4 1.790861 4 4c0 1.44275929-.763842 2.70711371-1.9091849 3.410722 1.6497664.7146943 2.7323812 2.3054378 3.4892659 4.3218837zm-5.580081 3.2673943c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9z" fill-rule="evenodd"/></symbol><symbol id="icon-alert" viewBox="0 0 18 18"><path d="m4 10h2.5c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-3.08578644l-1.12132034 1.1213203c-.18753638.1875364-.29289322.4418903-.29289322.7071068v.1715729h14v-.1715729c0-.2652165-.1053568-.5195704-.2928932-.7071068l-1.7071068-1.7071067v-3.4142136c0-2.76142375-2.2385763-5-5-5-2.76142375 0-5 2.23857625-5 5zm3 4c0 1.1045695.8954305 2 2 2s2-.8954305 2-2zm-5 0c-.55228475 0-1-.4477153-1-1v-.1715729c0-.530433.21071368-1.0391408.58578644-1.4142135l1.41421356-1.4142136v-3c0-3.3137085 2.6862915-6 6-6s6 2.6862915 6 6v3l1.4142136 1.4142136c.3750727.3750727.5857864.8837805.5857864 1.4142135v.1715729c0 .5522847-.4477153 1-1 1h-4c0 1.6568542-1.3431458 3-3 3-1.65685425 0-3-1.3431458-3-3z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-broad" viewBox="0 0 16 16"><path d="m6.10307866 2.97190702v7.69043288l2.44965196-2.44676915c.38776071-.38730439 1.0088052-.39493524 1.38498697-.01919617.38609051.38563612.38643641 1.01053024-.00013864 1.39665039l-4.12239817 4.11754683c-.38616704.3857126-1.01187344.3861062-1.39846576-.0000311l-4.12258206-4.11773056c-.38618426-.38572979-.39254614-1.00476697-.01636437-1.38050605.38609047-.38563611 1.01018509-.38751562 1.4012233.00306241l2.44985644 2.4469734v-8.67638639c0-.54139983.43698413-.98042709.98493125-.98159081l7.89910522-.0043627c.5451687 0 .9871152.44142642.9871152.98595351s-.4419465.98595351-.9871152.98595351z" fill-rule="evenodd" transform="matrix(-1 0 0 -1 14 15)"/></symbol><symbol id="icon-arrow-down" viewBox="0 0 16 16"><path d="m3.28337502 11.5302405 4.03074001 4.176208c.37758093.3912076.98937525.3916069 1.367372-.0000316l4.03091977-4.1763942c.3775978-.3912252.3838182-1.0190815.0160006-1.4001736-.3775061-.39113013-.9877245-.39303641-1.3700683.003106l-2.39538585 2.4818345v-11.6147896l-.00649339-.11662112c-.055753-.49733869-.46370161-.88337888-.95867408-.88337888-.49497246 0-.90292107.38604019-.95867408.88337888l-.00649338.11662112v11.6147896l-2.39518594-2.4816273c-.37913917-.39282218-.98637524-.40056175-1.35419292-.0194697-.37750607.3911302-.37784433 1.0249269.00013556 1.4165479z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-left" viewBox="0 0 16 16"><path d="m4.46975946 3.28337502-4.17620792 4.03074001c-.39120768.37758093-.39160691.98937525.0000316 1.367372l4.1763942 4.03091977c.39122514.3775978 1.01908149.3838182 1.40017357.0160006.39113012-.3775061.3930364-.9877245-.00310603-1.3700683l-2.48183446-2.39538585h11.61478958l.1166211-.00649339c.4973387-.055753.8833789-.46370161.8833789-.95867408 0-.49497246-.3860402-.90292107-.8833789-.95867408l-.1166211-.00649338h-11.61478958l2.4816273-2.39518594c.39282216-.37913917.40056173-.98637524.01946965-1.35419292-.39113012-.37750607-1.02492687-.37784433-1.41654791.00013556z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-right" viewBox="0 0 16 16"><path d="m11.5302405 12.716625 4.176208-4.03074003c.3912076-.37758093.3916069-.98937525-.0000316-1.367372l-4.1763942-4.03091981c-.3912252-.37759778-1.0190815-.38381821-1.4001736-.01600053-.39113013.37750607-.39303641.98772445.003106 1.37006824l2.4818345 2.39538588h-11.6147896l-.11662112.00649339c-.49733869.055753-.88337888.46370161-.88337888.95867408 0 .49497246.38604019.90292107.88337888.95867408l.11662112.00649338h11.6147896l-2.4816273 2.39518592c-.39282218.3791392-.40056175.9863753-.0194697 1.3541929.3911302.3775061 1.0249269.3778444 1.4165479-.0001355z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-sub" viewBox="0 0 16 16"><path d="m7.89692134 4.97190702v7.69043288l-2.44965196-2.4467692c-.38776071-.38730434-1.0088052-.39493519-1.38498697-.0191961-.38609047.3856361-.38643643 1.0105302.00013864 1.3966504l4.12239817 4.1175468c.38616704.3857126 1.01187344.3861062 1.39846576-.0000311l4.12258202-4.1177306c.3861843-.3857298.3925462-1.0047669.0163644-1.380506-.3860905-.38563612-1.0101851-.38751563-1.4012233.0030624l-2.44985643 2.4469734v-8.67638639c0-.54139983-.43698413-.98042709-.98493125-.98159081l-7.89910525-.0043627c-.54516866 0-.98711517.44142642-.98711517.98595351s.44194651.98595351.98711517.98595351z" fill-rule="evenodd"/></symbol><symbol id="icon-arrow-up" viewBox="0 0 16 16"><path d="m12.716625 4.46975946-4.03074003-4.17620792c-.37758093-.39120768-.98937525-.39160691-1.367372.0000316l-4.03091981 4.1763942c-.37759778.39122514-.38381821 1.01908149-.01600053 1.40017357.37750607.39113012.98772445.3930364 1.37006824-.00310603l2.39538588-2.48183446v11.61478958l.00649339.1166211c.055753.4973387.46370161.8833789.95867408.8833789.49497246 0 .90292107-.3860402.95867408-.8833789l.00649338-.1166211v-11.61478958l2.39518592 2.4816273c.3791392.39282216.9863753.40056173 1.3541929.01946965.3775061-.39113012.3778444-1.02492687-.0001355-1.41654791z" fill-rule="evenodd"/></symbol><symbol id="icon-article" viewBox="0 0 18 18"><path d="m13 15v-12.9906311c0-.0073595-.0019884-.0093689.0014977-.0093689l-11.00158888.00087166v13.00506804c0 .5482678.44615281.9940603.99415146.9940603h10.27350412c-.1701701-.2941734-.2675644-.6357129-.2675644-1zm-12 .0059397v-13.00506804c0-.5562408.44704472-1.00087166.99850233-1.00087166h11.00299537c.5510129 0 .9985023.45190985.9985023 1.0093689v2.9906311h3v9.9914698c0 1.1065798-.8927712 2.0085302-1.9940603 2.0085302h-12.01187942c-1.09954652 0-1.99406028-.8927712-1.99406028-1.9940603zm13-9.0059397v9c0 .5522847.4477153 1 1 1s1-.4477153 1-1v-9zm-10-2h7v4h-7zm1 1v2h5v-2zm-1 4h7v1h-7zm0 2h7v1h-7zm0 2h7v1h-7z" fill-rule="evenodd"/></symbol><symbol id="icon-audio" viewBox="0 0 18 18"><path d="m13.0957477 13.5588459c-.195279.1937043-.5119137.193729-.7072234.0000551-.1953098-.193674-.1953346-.5077061-.0000556-.7014104 1.0251004-1.0168342 1.6108711-2.3905226 1.6108711-3.85745208 0-1.46604976-.5850634-2.83898246-1.6090736-3.85566829-.1951894-.19379323-.1950192-.50782531.0003802-.70141028.1953993-.19358497.512034-.19341614.7072234.00037709 1.2094886 1.20083761 1.901635 2.8250555 1.901635 4.55670148 0 1.73268608-.6929822 3.35779608-1.9037571 4.55880738zm2.1233994 2.1025159c-.195234.193749-.5118687.1938462-.7072235.0002171-.1953548-.1936292-.1954528-.5076613-.0002189-.7014104 1.5832215-1.5711805 2.4881302-3.6939808 2.4881302-5.96012998 0-2.26581266-.9046382-4.3883241-2.487443-5.95944795-.1952117-.19377107-.1950777-.50780316.0002993-.70141031s.5120117-.19347426.7072234.00029682c1.7683321 1.75528196 2.7800854 4.12911258 2.7800854 6.66056144 0 2.53182498-1.0120556 4.90597838-2.7808529 6.66132328zm-14.21898205-3.6854911c-.5523759 0-1.00016505-.4441085-1.00016505-.991944v-3.96777631c0-.54783558.44778915-.99194407 1.00016505-.99194407h2.0003301l5.41965617-3.8393633c.44948677-.31842296 1.07413994-.21516983 1.39520191.23062232.12116339.16823446.18629727.36981184.18629727.57655577v12.01603479c0 .5478356-.44778914.9919441-1.00016505.9919441-.20845738 0-.41170538-.0645985-.58133413-.184766l-5.41965617-3.8393633zm0-.991944h2.32084805l5.68047235 4.0241292v-12.01603479l-5.68047235 4.02412928h-2.32084805z" fill-rule="evenodd"/></symbol><symbol id="icon-block" viewBox="0 0 24 24"><path d="m0 0h24v24h-24z" fill-rule="evenodd"/></symbol><symbol id="icon-book" viewBox="0 0 18 18"><path d="m4 13v-11h1v11h11v-11h-13c-.55228475 0-1 .44771525-1 1v10.2675644c.29417337-.1701701.63571286-.2675644 1-.2675644zm12 1h-13c-.55228475 0-1 .4477153-1 1s.44771525 1 1 1h13zm0 3h-13c-1.1045695 0-2-.8954305-2-2v-12c0-1.1045695.8954305-2 2-2h13c.5522847 0 1 .44771525 1 1v14c0 .5522847-.4477153 1-1 1zm-8.5-13h6c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-6c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm1 2h4c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-4c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-broad" viewBox="0 0 24 24"><path d="m9.18274226 7.81v7.7999954l2.48162734-2.4816273c.3928221-.3928221 1.0219731-.4005617 1.4030652-.0194696.3911301.3911301.3914806 1.0249268-.0001404 1.4165479l-4.17620796 4.1762079c-.39120769.3912077-1.02508144.3916069-1.41671995-.0000316l-4.1763942-4.1763942c-.39122514-.3912251-.39767006-1.0190815-.01657798-1.4001736.39113012-.3911301 1.02337106-.3930364 1.41951349.0031061l2.48183446 2.4818344v-8.7999954c0-.54911294.4426881-.99439484.99778758-.99557515l8.00221246-.00442485c.5522847 0 1 .44771525 1 1s-.4477153 1-1 1z" fill-rule="evenodd" transform="matrix(-1 0 0 -1 20.182742 24.805206)"/></symbol><symbol id="icon-calendar" viewBox="0 0 18 18"><path d="m12.5 0c.2761424 0 .5.21505737.5.49047852v.50952148h2c1.1072288 0 2 .89451376 2 2v12c0 1.1072288-.8945138 2-2 2h-12c-1.1072288 0-2-.8945138-2-2v-12c0-1.1072288.89451376-2 2-2h1v1h-1c-.55393837 0-1 .44579254-1 1v3h14v-3c0-.55393837-.4457925-1-1-1h-2v1.50952148c0 .27088381-.2319336.49047852-.5.49047852-.2761424 0-.5-.21505737-.5-.49047852v-3.01904296c0-.27088381.2319336-.49047852.5-.49047852zm3.5 7h-14v8c0 .5539384.44579254 1 1 1h12c.5539384 0 1-.4457925 1-1zm-11 6v1h-1v-1zm3 0v1h-1v-1zm3 0v1h-1v-1zm-6-2v1h-1v-1zm3 0v1h-1v-1zm6 0v1h-1v-1zm-3 0v1h-1v-1zm-3-2v1h-1v-1zm6 0v1h-1v-1zm-3 0v1h-1v-1zm-5.5-9c.27614237 0 .5.21505737.5.49047852v.50952148h5v1h-5v1.50952148c0 .27088381-.23193359.49047852-.5.49047852-.27614237 0-.5-.21505737-.5-.49047852v-3.01904296c0-.27088381.23193359-.49047852.5-.49047852z" fill-rule="evenodd"/></symbol><symbol id="icon-cart" viewBox="0 0 18 18"><path d="m5 14c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm10 0c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm-10 1c-.55228475 0-1 .4477153-1 1s.44771525 1 1 1 1-.4477153 1-1-.44771525-1-1-1zm10 0c-.5522847 0-1 .4477153-1 1s.4477153 1 1 1 1-.4477153 1-1-.4477153-1-1-1zm-12.82032249-15c.47691417 0 .88746157.33678127.98070211.80449199l.23823144 1.19501025 13.36277974.00045554c.5522847.00001882.9999659.44774934.9999659 1.00004222 0 .07084994-.0075361.14150708-.022474.2107727l-1.2908094 5.98534344c-.1007861.46742419-.5432548.80388386-1.0571651.80388386h-10.24805106c-.59173366 0-1.07142857.4477153-1.07142857 1 0 .5128358.41361449.9355072.94647737.9932723l.1249512.0067277h10.35933776c.2749512 0 .4979349.2228539.4979349.4978051 0 .2749417-.2227336.4978951-.4976753.4980063l-10.35959736.0041886c-1.18346732 0-2.14285714-.8954305-2.14285714-2 0-.6625717.34520317-1.24989198.87690425-1.61383592l-1.63768102-8.19004794c-.01312273-.06561364-.01950005-.131011-.0196107-.19547395l-1.71961253-.00064219c-.27614237 0-.5-.22385762-.5-.5 0-.27614237.22385763-.5.5-.5zm14.53193359 2.99950224h-13.11300004l1.20580469 6.02530174c.11024034-.0163252.22327998-.02480398.33844139-.02480398h10.27064786z"/></symbol><symbol id="icon-chevron-less" viewBox="0 0 10 10"><path d="m5.58578644 4-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4c-.39052429.39052429-1.02368927.39052429-1.41421356 0s-.39052429-1.02368927 0-1.41421356z" fill-rule="evenodd" transform="matrix(0 -1 -1 0 9 9)"/></symbol><symbol id="icon-chevron-more" viewBox="0 0 10 10"><path d="m5.58578644 6-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4.00000002c-.39052429.3905243-1.02368927.3905243-1.41421356 0s-.39052429-1.02368929 0-1.41421358z" fill-rule="evenodd" transform="matrix(0 1 -1 0 11 1)"/></symbol><symbol id="icon-chevron-right" viewBox="0 0 10 10"><path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/></symbol><symbol id="icon-circle-fill" viewBox="0 0 16 16"><path d="m8 14c-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6 6 2.6862915 6 6-2.6862915 6-6 6z" fill-rule="evenodd"/></symbol><symbol id="icon-circle" viewBox="0 0 16 16"><path d="m8 12c2.209139 0 4-1.790861 4-4s-1.790861-4-4-4-4 1.790861-4 4 1.790861 4 4 4zm0 2c-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6 6 2.6862915 6 6-2.6862915 6-6 6z" fill-rule="evenodd"/></symbol><symbol id="icon-citation" viewBox="0 0 18 18"><path d="m8.63593473 5.99995183c2.20913897 0 3.99999997 1.79084375 3.99999997 3.99996146 0 1.40730761-.7267788 2.64486871-1.8254829 3.35783281 1.6240224.6764218 2.8754442 2.0093871 3.4610603 3.6412466l-1.0763845.000006c-.5310008-1.2078237-1.5108121-2.1940153-2.7691712-2.7181346l-.79002167-.329052v-1.023992l.63016577-.4089232c.8482885-.5504661 1.3698342-1.4895187 1.3698342-2.51898361 0-1.65683828-1.3431457-2.99996146-2.99999997-2.99996146-1.65685425 0-3 1.34312318-3 2.99996146 0 1.02946491.52154569 1.96851751 1.36983419 2.51898361l.63016581.4089232v1.023992l-.79002171.329052c-1.25835905.5241193-2.23817037 1.5103109-2.76917113 2.7181346l-1.07638453-.000006c.58561612-1.6318595 1.8370379-2.9648248 3.46106024-3.6412466-1.09870405-.7129641-1.82548287-1.9505252-1.82548287-3.35783281 0-2.20911771 1.790861-3.99996146 4-3.99996146zm7.36897597-4.99995183c1.1018574 0 1.9950893.89353404 1.9950893 2.00274083v5.994422c0 1.10608317-.8926228 2.00274087-1.9950893 2.00274087l-3.0049107-.0009037v-1l3.0049107.00091329c.5490631 0 .9950893-.44783123.9950893-1.00275046v-5.994422c0-.55646537-.4450595-1.00275046-.9950893-1.00275046h-14.00982141c-.54906309 0-.99508929.44783123-.99508929 1.00275046v5.9971821c0 .66666024.33333333.99999036 1 .99999036l2-.00091329v1l-2 .0009037c-1 0-2-.99999041-2-1.99998077v-5.9971821c0-1.10608322.8926228-2.00274083 1.99508929-2.00274083zm-8.5049107 2.9999711c.27614237 0 .5.22385547.5.5 0 .2761349-.22385763.5-.5.5h-4c-.27614237 0-.5-.2238651-.5-.5 0-.27614453.22385763-.5.5-.5zm3 0c.2761424 0 .5.22385547.5.5 0 .2761349-.2238576.5-.5.5h-1c-.27614237 0-.5-.2238651-.5-.5 0-.27614453.22385763-.5.5-.5zm4 0c.2761424 0 .5.22385547.5.5 0 .2761349-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238651-.5-.5 0-.27614453.2238576-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-close" viewBox="0 0 16 16"><path d="m2.29679575 12.2772478c-.39658757.3965876-.39438847 1.0328109-.00062148 1.4265779.39651227.3965123 1.03246768.3934888 1.42657791-.0006214l4.27724782-4.27724787 4.2772478 4.27724787c.3965876.3965875 1.0328109.3943884 1.4265779.0006214.3965123-.3965122.3934888-1.0324677-.0006214-1.4265779l-4.27724787-4.2772478 4.27724787-4.27724782c.3965875-.39658757.3943884-1.03281091.0006214-1.42657791-.3965122-.39651226-1.0324677-.39348875-1.4265779.00062148l-4.2772478 4.27724782-4.27724782-4.27724782c-.39658757-.39658757-1.03281091-.39438847-1.42657791-.00062148-.39651226.39651227-.39348875 1.03246768.00062148 1.42657791l4.27724782 4.27724782z" fill-rule="evenodd"/></symbol><symbol id="icon-collections" viewBox="0 0 18 18"><path d="m15 4c1.1045695 0 2 .8954305 2 2v9c0 1.1045695-.8954305 2-2 2h-8c-1.1045695 0-2-.8954305-2-2h1c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h8c.5128358 0 .9355072-.3860402.9932723-.8833789l.0067277-.1166211v-9c0-.51283584-.3860402-.93550716-.8833789-.99327227l-.1166211-.00672773h-1v-1zm-4-3c1.1045695 0 2 .8954305 2 2v9c0 1.1045695-.8954305 2-2 2h-8c-1.1045695 0-2-.8954305-2-2v-9c0-1.1045695.8954305-2 2-2zm0 1h-8c-.51283584 0-.93550716.38604019-.99327227.88337887l-.00672773.11662113v9c0 .5128358.38604019.9355072.88337887.9932723l.11662113.0067277h8c.5128358 0 .9355072-.3860402.9932723-.8833789l.0067277-.1166211v-9c0-.51283584-.3860402-.93550716-.8833789-.99327227zm-1.5 7c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm0-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm0-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-5c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-compare" viewBox="0 0 18 18"><path d="m12 3c3.3137085 0 6 2.6862915 6 6s-2.6862915 6-6 6c-1.0928452 0-2.11744941-.2921742-2.99996061-.8026704-.88181407.5102749-1.90678042.8026704-3.00003939.8026704-3.3137085 0-6-2.6862915-6-6s2.6862915-6 6-6c1.09325897 0 2.11822532.29239547 3.00096303.80325037.88158756-.51107621 1.90619177-.80325037 2.99903697-.80325037zm-6 1c-2.76142375 0-5 2.23857625-5 5 0 2.7614237 2.23857625 5 5 5 .74397391 0 1.44999672-.162488 2.08451611-.4539116-1.27652344-1.1000812-2.08451611-2.7287264-2.08451611-4.5460884s.80799267-3.44600721 2.08434391-4.5463015c-.63434719-.29121054-1.34037-.4536985-2.08434391-.4536985zm6 0c-.7439739 0-1.4499967.16248796-2.08451611.45391156 1.27652341 1.10008123 2.08451611 2.72872644 2.08451611 4.54608844s-.8079927 3.4460072-2.08434391 4.5463015c.63434721.2912105 1.34037001.4536985 2.08434391.4536985 2.7614237 0 5-2.2385763 5-5 0-2.76142375-2.2385763-5-5-5zm-1.4162763 7.0005324h-3.16744736c.15614659.3572676.35283837.6927622.58425872 1.0006671h1.99892988c.23142036-.3079049.42811216-.6433995.58425876-1.0006671zm.4162763-2.0005324h-4c0 .34288501.0345146.67770871.10025909 1.0011864h3.79948181c.0657445-.32347769.1002591-.65830139.1002591-1.0011864zm-.4158423-1.99953894h-3.16831543c-.13859957.31730812-.24521946.651783-.31578599.99935097h3.79988742c-.0705665-.34756797-.1771864-.68204285-.315786-.99935097zm-1.58295822-1.999926-.08316107.06199199c-.34550042.27081213-.65446126.58611297-.91825862.93727862h2.00044041c-.28418626-.37830727-.6207872-.71499149-.99902072-.99927061z" fill-rule="evenodd"/></symbol><symbol id="icon-download-file" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm0 1h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v14.00982141c0 .5500396.44491393.9950893.99406028.9950893h12.01187942c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717zm-1.5046024 4c.27614237 0 .5.21637201.5.49209595v6.14827645l1.7462789-1.77990922c.1933927-.1971171.5125222-.19455839.7001689-.0069117.1932998.19329992.1910058.50899492-.0027774.70277812l-2.59089271 2.5908927c-.19483374.1948337-.51177825.1937771-.70556873-.0000133l-2.59099079-2.5909908c-.19484111-.1948411-.19043735-.5151448-.00279066-.70279146.19329987-.19329987.50465175-.19237083.70018565.00692852l1.74638684 1.78001764v-6.14827695c0-.27177709.23193359-.49209595.5-.49209595z" fill-rule="evenodd"/></symbol><symbol id="icon-download" viewBox="0 0 16 16"><path d="m12.9975267 12.999368c.5467123 0 1.0024733.4478567 1.0024733 1.000316 0 .5563109-.4488226 1.000316-1.0024733 1.000316h-9.99505341c-.54671233 0-1.00247329-.4478567-1.00247329-1.000316 0-.5563109.44882258-1.000316 1.00247329-1.000316zm-4.9975267-11.999368c.55228475 0 1 .44497754 1 .99589209v6.80214418l2.4816273-2.48241149c.3928222-.39294628 1.0219732-.4006883 1.4030652-.01947579.3911302.39125371.3914806 1.02525073-.0001404 1.41699553l-4.17620792 4.17752758c-.39120769.3913313-1.02508144.3917306-1.41671995-.0000316l-4.17639421-4.17771394c-.39122513-.39134876-.39767006-1.01940351-.01657797-1.40061601.39113012-.39125372 1.02337105-.3931606 1.41951349.00310701l2.48183446 2.48261871v-6.80214418c0-.55001601.44386482-.99589209 1-.99589209z" fill-rule="evenodd"/></symbol><symbol id="icon-editors" viewBox="0 0 18 18"><path d="m8.72592184 2.54588137c-.48811714-.34391207-1.08343326-.54588137-1.72592184-.54588137-1.65685425 0-3 1.34314575-3 3 0 1.02947485.5215457 1.96853646 1.3698342 2.51900785l.6301658.40892721v1.02400182l-.79002171.32905522c-1.93395773.8055207-3.20997829 2.7024791-3.20997829 4.8180274v.9009805h-1v-.9009805c0-2.5479714 1.54557359-4.79153984 3.82548288-5.7411543-1.09870406-.71297106-1.82548288-1.95054399-1.82548288-3.3578652 0-2.209139 1.790861-4 4-4 1.09079823 0 2.07961816.43662103 2.80122451 1.1446278-.37707584.09278571-.7373238.22835063-1.07530267.40125357zm-2.72592184 14.45411863h-1v-.9009805c0-2.5479714 1.54557359-4.7915398 3.82548288-5.7411543-1.09870406-.71297106-1.82548288-1.95054399-1.82548288-3.3578652 0-2.209139 1.790861-4 4-4s4 1.790861 4 4c0 1.40732121-.7267788 2.64489414-1.8254829 3.3578652 2.2799093.9496145 3.8254829 3.1931829 3.8254829 5.7411543v.9009805h-1v-.9009805c0-2.1155483-1.2760206-4.0125067-3.2099783-4.8180274l-.7900217-.3290552v-1.02400184l.6301658-.40892721c.8482885-.55047139 1.3698342-1.489533 1.3698342-2.51900785 0-1.65685425-1.3431458-3-3-3-1.65685425 0-3 1.34314575-3 3 0 1.02947485.5215457 1.96853646 1.3698342 2.51900785l.6301658.40892721v1.02400184l-.79002171.3290552c-1.93395773.8055207-3.20997829 2.7024791-3.20997829 4.8180274z" fill-rule="evenodd"/></symbol><symbol id="icon-email" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587h-14.00982141c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm0 1h-14.00982141c-.54871518 0-.99508929.44887827-.99508929 1.00585866v9.98828264c0 .5572961.44630695 1.0058587.99508929 1.0058587h14.00982141c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-.0049107 2.55749512v1.44250488l-7 4-7-4v-1.44250488l7 4z" fill-rule="evenodd"/></symbol><symbol id="icon-error" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm2.8630343 4.71100931-2.8630343 2.86303426-2.86303426-2.86303426c-.39658757-.39658757-1.03281091-.39438847-1.4265779-.00062147-.39651227.39651226-.39348876 1.03246767.00062147 1.4265779l2.86303426 2.86303426-2.86303426 2.8630343c-.39658757.3965875-.39438847 1.0328109-.00062147 1.4265779.39651226.3965122 1.03246767.3934887 1.4265779-.0006215l2.86303426-2.8630343 2.8630343 2.8630343c.3965875.3965876 1.0328109.3943885 1.4265779.0006215.3965122-.3965123.3934887-1.0324677-.0006215-1.4265779l-2.8630343-2.8630343 2.8630343-2.86303426c.3965876-.39658757.3943885-1.03281091.0006215-1.4265779-.3965123-.39651227-1.0324677-.39348876-1.4265779.00062147z" fill-rule="evenodd"/></symbol><symbol id="icon-ethics" viewBox="0 0 18 18"><path d="m6.76384967 1.41421356.83301651-.8330165c.77492941-.77492941 2.03133823-.77492941 2.80626762 0l.8330165.8330165c.3750728.37507276.8837806.58578644 1.4142136.58578644h1.3496361c1.1045695 0 2 .8954305 2 2v1.34963611c0 .53043298.2107137 1.03914081.5857864 1.41421356l.8330165.83301651c.7749295.77492941.7749295 2.03133823 0 2.80626762l-.8330165.8330165c-.3750727.3750728-.5857864.8837806-.5857864 1.4142136v1.3496361c0 1.1045695-.8954305 2-2 2h-1.3496361c-.530433 0-1.0391408.2107137-1.4142136.5857864l-.8330165.8330165c-.77492939.7749295-2.03133821.7749295-2.80626762 0l-.83301651-.8330165c-.37507275-.3750727-.88378058-.5857864-1.41421356-.5857864h-1.34963611c-1.1045695 0-2-.8954305-2-2v-1.3496361c0-.530433-.21071368-1.0391408-.58578644-1.4142136l-.8330165-.8330165c-.77492941-.77492939-.77492941-2.03133821 0-2.80626762l.8330165-.83301651c.37507276-.37507275.58578644-.88378058.58578644-1.41421356v-1.34963611c0-1.1045695.8954305-2 2-2h1.34963611c.53043298 0 1.03914081-.21071368 1.41421356-.58578644zm-1.41421356 1.58578644h-1.34963611c-.55228475 0-1 .44771525-1 1v1.34963611c0 .79564947-.31607052 1.55871121-.87867966 2.12132034l-.8330165.83301651c-.38440512.38440512-.38440512 1.00764896 0 1.39205408l.8330165.83301646c.56260914.5626092.87867966 1.3256709.87867966 2.1213204v1.3496361c0 .5522847.44771525 1 1 1h1.34963611c.79564947 0 1.55871121.3160705 2.12132034.8786797l.83301651.8330165c.38440512.3844051 1.00764896.3844051 1.39205408 0l.83301646-.8330165c.5626092-.5626092 1.3256709-.8786797 2.1213204-.8786797h1.3496361c.5522847 0 1-.4477153 1-1v-1.3496361c0-.7956495.3160705-1.5587112.8786797-2.1213204l.8330165-.83301646c.3844051-.38440512.3844051-1.00764896 0-1.39205408l-.8330165-.83301651c-.5626092-.56260913-.8786797-1.32567087-.8786797-2.12132034v-1.34963611c0-.55228475-.4477153-1-1-1h-1.3496361c-.7956495 0-1.5587112-.31607052-2.1213204-.87867966l-.83301646-.8330165c-.38440512-.38440512-1.00764896-.38440512-1.39205408 0l-.83301651.8330165c-.56260913.56260914-1.32567087.87867966-2.12132034.87867966zm3.58698944 11.4960218c-.02081224.002155-.04199226.0030286-.06345763.002542-.98766446-.0223875-1.93408568-.3063547-2.75885125-.8155622-.23496767-.1450683-.30784554-.4531483-.16277726-.688116.14506827-.2349677.45314827-.3078455.68811595-.1627773.67447084.4164161 1.44758575.6483839 2.25617384.6667123.01759529.0003988.03495764.0017019.05204365.0038639.01713363-.0017748.03452416-.0026845.05212715-.0026845 2.4852814 0 4.5-2.0147186 4.5-4.5 0-1.04888973-.3593547-2.04134635-1.0074477-2.83787157-.1742817-.21419731-.1419238-.5291218.0722736-.70340353.2141973-.17428173.5291218-.14192375.7034035.07227357.7919032.97327203 1.2317706 2.18808682 1.2317706 3.46900153 0 3.0375661-2.4624339 5.5-5.5 5.5-.02146768 0-.04261937-.0013529-.06337445-.0039782zm1.57975095-10.78419583c.2654788.07599731.419084.35281842.3430867.61829728-.0759973.26547885-.3528185.419084-.6182973.3430867-.37560116-.10752146-.76586237-.16587951-1.15568824-.17249193-2.5587807-.00064534-4.58547766 2.00216524-4.58547766 4.49928198 0 .62691557.12797645 1.23496.37274865 1.7964426.11035133.2531347-.0053975.5477984-.25853224.6581497-.25313473.1103514-.54779841-.0053975-.65814974-.2585322-.29947131-.6869568-.45606667-1.43097603-.45606667-2.1960601 0-3.05211432 2.47714695-5.50006595 5.59399617-5.49921198.48576182.00815502.96289603.0795037 1.42238033.21103795zm-1.9766658 6.41091303 2.69835-2.94655317c.1788432-.21040373.4943901-.23598862.7047939-.05714545.2104037.17884318.2359886.49439014.0571454.70479387l-3.01637681 3.34277395c-.18039088.1999106-.48669547.2210637-.69285412.0478478l-1.93095347-1.62240047c-.21213845-.17678204-.24080048-.49206439-.06401844-.70420284.17678204-.21213844.49206439-.24080048.70420284-.06401844z" fill-rule="evenodd"/></symbol><symbol id="icon-expand"><path d="M7.498 11.918a.997.997 0 0 0-.003-1.411.995.995 0 0 0-1.412-.003l-4.102 4.102v-3.51A1 1 0 0 0 .98 10.09.992.992 0 0 0 0 11.092V17c0 .554.448 1.002 1.002 1.002h5.907c.554 0 1.002-.45 1.002-1.003 0-.539-.45-.978-1.006-.978h-3.51zm3.005-5.835a.997.997 0 0 0 .003 1.412.995.995 0 0 0 1.411.003l4.103-4.103v3.51a1 1 0 0 0 1.001 1.006A.992.992 0 0 0 18 6.91V1.002A1 1 0 0 0 17 0h-5.907a1.003 1.003 0 0 0-1.002 1.003c0 .539.45.978 1.006.978h3.51z" fill-rule="evenodd"/></symbol><symbol id="icon-explore" viewBox="0 0 18 18"><path d="m9 17c4.418278 0 8-3.581722 8-8s-3.581722-8-8-8-8 3.581722-8 8 3.581722 8 8 8zm0 1c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9zm0-2.5c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5c2.969509 0 5.400504-2.3575119 5.497023-5.31714844.0090007-.27599565.2400359-.49243782.5160315-.48343711.2759957.0090007.4924378.2400359.4834371.51603155-.114093 3.4985237-2.9869632 6.284554-6.4964916 6.284554zm-.29090657-12.99359748c.27587424-.01216621.50937715.20161139.52154336.47748563.01216621.27587423-.20161139.50937715-.47748563.52154336-2.93195733.12930094-5.25315116 2.54886451-5.25315116 5.49456849 0 .27614237-.22385763.5-.5.5s-.5-.22385763-.5-.5c0-3.48142406 2.74307146-6.34074398 6.20909343-6.49359748zm1.13784138 8.04763908-1.2004882-1.20048821c-.19526215-.19526215-.19526215-.51184463 0-.70710678s.51184463-.19526215.70710678 0l1.20048821 1.2004882 1.6006509-4.00162734-4.50670359 1.80268144-1.80268144 4.50670359zm4.10281269-6.50378907-2.6692597 6.67314927c-.1016411.2541026-.3029834.4554449-.557086.557086l-6.67314927 2.6692597 2.66925969-6.67314926c.10164107-.25410266.30298336-.45544495.55708602-.55708602z" fill-rule="evenodd"/></symbol><symbol id="icon-filter" viewBox="0 0 16 16"><path d="m14.9738641 0c.5667192 0 1.0261359.4477136 1.0261359 1 0 .24221858-.0902161.47620768-.2538899.65849851l-5.6938314 6.34147206v5.49997973c0 .3147562-.1520673.6111434-.4104543.7999971l-2.05227171 1.4999945c-.45337535.3313696-1.09655869.2418269-1.4365902-.1999993-.13321514-.1730955-.20522717-.3836284-.20522717-.5999978v-6.99997423l-5.69383133-6.34147206c-.3731872-.41563511-.32996891-1.0473954.09653074-1.41107611.18705584-.15950448.42716133-.2474224.67571519-.2474224zm-5.9218641 8.5h-2.105v6.491l.01238459.0070843.02053271.0015705.01955278-.0070558 2.0532976-1.4990996zm-8.02585008-7.5-.01564945.00240169 5.83249953 6.49759831h2.313l5.836-6.499z"/></symbol><symbol id="icon-home" viewBox="0 0 18 18"><path d="m9 5-6 6v5h4v-4h4v4h4v-5zm7 6.5857864v4.4142136c0 .5522847-.4477153 1-1 1h-5v-4h-2v4h-5c-.55228475 0-1-.4477153-1-1v-4.4142136c-.25592232 0-.51184464-.097631-.70710678-.2928932l-.58578644-.5857864c-.39052429-.3905243-.39052429-1.02368929 0-1.41421358l8.29289322-8.29289322 8.2928932 8.29289322c.3905243.39052429.3905243 1.02368928 0 1.41421358l-.5857864.5857864c-.1952622.1952622-.4511845.2928932-.7071068.2928932zm-7-9.17157284-7.58578644 7.58578644.58578644.5857864 7-6.99999996 7 6.99999996.5857864-.5857864z" fill-rule="evenodd"/></symbol><symbol id="icon-image" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm-3.49645283 10.1752453-3.89407257 6.7495552c.11705545.048464.24538859.0751995.37998328.0751995h10.60290092l-2.4329715-4.2154691-1.57494129 2.7288098zm8.49779013 6.8247547c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v13.98991071l4.50814957-7.81026689 3.08089884 5.33809539 1.57494129-2.7288097 3.5875735 6.2159812zm-3.0059397-11c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2-2-.8954305-2-2 .8954305-2 2-2zm0 1c-.5522847 0-1 .44771525-1 1s.4477153 1 1 1 1-.44771525 1-1-.4477153-1-1-1z" fill-rule="evenodd"/></symbol><symbol id="icon-info" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm0 7h-1.5l-.11662113.00672773c-.49733868.05776511-.88337887.48043643-.88337887.99327227 0 .47338693.32893365.86994729.77070917.97358929l.1126697.01968298.11662113.00672773h.5v3h-.5l-.11662113.0067277c-.42082504.0488782-.76196299.3590206-.85696816.7639815l-.01968298.1126697-.00672773.1166211.00672773.1166211c.04887817.4208251.35902055.761963.76398144.8569682l.1126697.019683.11662113.0067277h3l.1166211-.0067277c.4973387-.0577651.8833789-.4804365.8833789-.9932723 0-.4733869-.3289337-.8699473-.7707092-.9735893l-.1126697-.019683-.1166211-.0067277h-.5v-4l-.00672773-.11662113c-.04887817-.42082504-.35902055-.76196299-.76398144-.85696816l-.1126697-.01968298zm0-3.25c-.69035594 0-1.25.55964406-1.25 1.25s.55964406 1.25 1.25 1.25 1.25-.55964406 1.25-1.25-.55964406-1.25-1.25-1.25z" fill-rule="evenodd"/></symbol><symbol id="icon-institution" viewBox="0 0 18 18"><path d="m7 16.9998189v-2.0003623h4v2.0003623h2v-3.0005434h-8v3.0005434zm-3-10.00181122h-1.52632364c-.27614237 0-.5-.22389817-.5-.50009056 0-.13995446.05863589-.27350497.16166338-.36820841l1.23156713-1.13206327h-2.36690687v12.00217346h3v-2.0003623h-3v-1.0001811h3v-1.0001811h1v-4.00072448h-1zm10 0v2.00036224h-1v4.00072448h1v1.0001811h3v1.0001811h-3v2.0003623h3v-12.00217346h-2.3695309l1.2315671 1.13206327c.2033191.186892.2166633.50325042.0298051.70660631-.0946863.10304615-.2282126.16169266-.3681417.16169266zm3-3.00054336c.5522847 0 1 .44779634 1 1.00018112v13.00235456h-18v-13.00235456c0-.55238478.44771525-1.00018112 1-1.00018112h3.45499992l4.20535144-3.86558216c.19129876-.17584288.48537447-.17584288.67667324 0l4.2053514 3.86558216zm-4 3.00054336h-8v1.00018112h8zm-2 6.00108672h1v-4.00072448h-1zm-1 0v-4.00072448h-2v4.00072448zm-3 0v-4.00072448h-1v4.00072448zm8-4.00072448c.5522847 0 1 .44779634 1 1.00018112v2.00036226h-2v-2.00036226c0-.55238478.4477153-1.00018112 1-1.00018112zm-12 0c.55228475 0 1 .44779634 1 1.00018112v2.00036226h-2v-2.00036226c0-.55238478.44771525-1.00018112 1-1.00018112zm5.99868798-7.81907007-5.24205601 4.81852671h10.48411203zm.00131202 3.81834559c-.55228475 0-1-.44779634-1-1.00018112s.44771525-1.00018112 1-1.00018112 1 .44779634 1 1.00018112-.44771525 1.00018112-1 1.00018112zm-1 11.00199236v1.0001811h2v-1.0001811z" fill-rule="evenodd"/></symbol><symbol id="icon-location" viewBox="0 0 18 18"><path d="m9.39521328 16.2688008c.79596342-.7770119 1.59208152-1.6299956 2.33285652-2.5295081 1.4020032-1.7024324 2.4323601-3.3624519 2.9354918-4.871847.2228715-.66861448.3364384-1.29323246.3364384-1.8674457 0-3.3137085-2.6862915-6-6-6-3.36356866 0-6 2.60156856-6 6 0 .57421324.11356691 1.19883122.3364384 1.8674457.50313169 1.5093951 1.53348863 3.1694146 2.93549184 4.871847.74077492.8995125 1.53689309 1.7524962 2.33285648 2.5295081.13694479.1336842.26895677.2602648.39521328.3793207.12625651-.1190559.25826849-.2456365.39521328-.3793207zm-.39521328 1.7311992s-7-6-7-11c0-4 3.13400675-7 7-7 3.8659932 0 7 3.13400675 7 7 0 5-7 11-7 11zm0-8c-1.65685425 0-3-1.34314575-3-3s1.34314575-3 3-3c1.6568542 0 3 1.34314575 3 3s-1.3431458 3-3 3zm0-1c1.1045695 0 2-.8954305 2-2s-.8954305-2-2-2-2 .8954305-2 2 .8954305 2 2 2z" fill-rule="evenodd"/></symbol><symbol id="icon-minus" viewBox="0 0 16 16"><path d="m2.00087166 7h11.99825664c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-11.99825664c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-newsletter" viewBox="0 0 18 18"><path d="m9 11.8482489 2-1.1428571v-1.7053918h-4v1.7053918zm-3-1.7142857v-2.1339632h6v2.1339632l3-1.71428574v-6.41967746h-12v6.41967746zm10-5.3839632 1.5299989.95624934c.2923814.18273835.4700011.50320827.4700011.8479983v8.44575236c0 1.1045695-.8954305 2-2 2h-14c-1.1045695 0-2-.8954305-2-2v-8.44575236c0-.34479003.1776197-.66525995.47000106-.8479983l1.52999894-.95624934v-2.75c0-.55228475.44771525-1 1-1h12c.5522847 0 1 .44771525 1 1zm0 1.17924764v3.07075236l-7 4-7-4v-3.07075236l-1 .625v8.44575236c0 .5522847.44771525 1 1 1h14c.5522847 0 1-.4477153 1-1v-8.44575236zm-10-1.92924764h6v1h-6zm-1 2h8v1h-8z" fill-rule="evenodd"/></symbol><symbol id="icon-orcid" viewBox="0 0 18 18"><path d="m9 1c4.418278 0 8 3.581722 8 8s-3.581722 8-8 8-8-3.581722-8-8 3.581722-8 8-8zm-2.90107518 5.2732337h-1.41865256v7.1712107h1.41865256zm4.55867178.02508949h-2.99247027v7.14612121h2.91062487c.7673039 0 1.4476365-.1483432 2.0410182-.445034s1.0511995-.7152915 1.3734671-1.2558144c.3222677-.540523.4833991-1.1603247.4833991-1.85942385 0-.68545815-.1602789-1.30270225-.4808414-1.85175082-.3205625-.54904856-.7707074-.97532211-1.3504481-1.27883343-.5797408-.30351132-1.2413173-.45526471-1.9847495-.45526471zm-.1892674 1.07933542c.7877654 0 1.4143875.22336734 1.8798852.67010873.4654977.44674138.698243 1.05546001.698243 1.82617415 0 .74343221-.2310402 1.34447791-.6931277 1.80315511-.4620874.4586773-1.0750688.6880124-1.8389625.6880124h-1.46810075v-4.98745039zm-5.08652545-3.71099194c-.21825533 0-.410525.08444276-.57681478.25333081-.16628977.16888806-.24943341.36245684-.24943341.58071218 0 .22345188.08314364.41961891.24943341.58850696.16628978.16888806.35855945.25333082.57681478.25333082.233845 0 .43390938-.08314364.60019916-.24943342.16628978-.16628977.24943342-.36375592.24943342-.59240436 0-.233845-.08314364-.43131115-.24943342-.59240437s-.36635416-.24163862-.60019916-.24163862z" fill-rule="evenodd"/></symbol><symbol id="icon-plus" viewBox="0 0 16 16"><path d="m2.00087166 7h4.99912834v-4.99912834c0-.55276616.44386482-1.00087166 1-1.00087166.55228475 0 1 .44463086 1 1.00087166v4.99912834h4.9991283c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-4.9991283v4.9991283c0 .5527662-.44386482 1.0008717-1 1.0008717-.55228475 0-1-.4446309-1-1.0008717v-4.9991283h-4.99912834c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-print" viewBox="0 0 18 18"><path d="m16.0049107 5h-14.00982141c-.54941618 0-.99508929.4467783-.99508929.99961498v6.00077002c0 .5570958.44271433.999615.99508929.999615h1.00491071v-3h12v3h1.0049107c.5494162 0 .9950893-.4467783.9950893-.999615v-6.00077002c0-.55709576-.4427143-.99961498-.9950893-.99961498zm-2.0049107-1v-2.00208688c0-.54777062-.4519464-.99791312-1.0085302-.99791312h-7.9829396c-.55661731 0-1.0085302.44910695-1.0085302.99791312v2.00208688zm1 10v2.0018986c0 1.103521-.9019504 1.9981014-2.0085302 1.9981014h-7.9829396c-1.1092806 0-2.0085302-.8867064-2.0085302-1.9981014v-2.0018986h-1.00491071c-1.10185739 0-1.99508929-.8874333-1.99508929-1.999615v-6.00077002c0-1.10435686.8926228-1.99961498 1.99508929-1.99961498h1.00491071v-2.00208688c0-1.10341695.90195036-1.99791312 2.0085302-1.99791312h7.9829396c1.1092806 0 2.0085302.89826062 2.0085302 1.99791312v2.00208688h1.0049107c1.1018574 0 1.9950893.88743329 1.9950893 1.99961498v6.00077002c0 1.1043569-.8926228 1.999615-1.9950893 1.999615zm-1-3h-10v5.0018986c0 .5546075.44702548.9981014 1.0085302.9981014h7.9829396c.5565964 0 1.0085302-.4491701 1.0085302-.9981014zm-9 1h8v1h-8zm0 2h5v1h-5zm9-5c-.5522847 0-1-.44771525-1-1s.4477153-1 1-1 1 .44771525 1 1-.4477153 1-1 1z" fill-rule="evenodd"/></symbol><symbol id="icon-search" viewBox="0 0 22 22"><path d="M21.697 20.261a1.028 1.028 0 01.01 1.448 1.034 1.034 0 01-1.448-.01l-4.267-4.267A9.812 9.811 0 010 9.812a9.812 9.811 0 1117.43 6.182zM9.812 18.222A8.41 8.41 0 109.81 1.403a8.41 8.41 0 000 16.82z" fill-rule="evenodd"/></symbol><symbol id="icon-social-facebook" viewBox="0 0 24 24"><path d="m6.00368507 20c-1.10660471 0-2.00368507-.8945138-2.00368507-1.9940603v-12.01187942c0-1.10128908.89451376-1.99406028 1.99406028-1.99406028h12.01187942c1.1012891 0 1.9940603.89451376 1.9940603 1.99406028v12.01187942c0 1.1012891-.88679 1.9940603-2.0032184 1.9940603h-2.9570132v-6.1960818h2.0797387l.3114113-2.414723h-2.39115v-1.54164807c0-.69911803.1941355-1.1755439 1.1966615-1.1755439l1.2786739-.00055875v-2.15974763l-.2339477-.02492088c-.3441234-.03134957-.9500153-.07025255-1.6293054-.07025255-1.8435726 0-3.1057323 1.12531866-3.1057323 3.19187953v1.78079225h-2.0850778v2.414723h2.0850778v6.1960818z" fill-rule="evenodd"/></symbol><symbol id="icon-social-twitter" viewBox="0 0 24 24"><path d="m18.8767135 6.87445248c.7638174-.46908424 1.351611-1.21167363 1.6250764-2.09636345-.7135248.43394112-1.50406.74870123-2.3464594.91677702-.6695189-.73342162-1.6297913-1.19486605-2.6922204-1.19486605-2.0399895 0-3.6933555 1.69603749-3.6933555 3.78628909 0 .29642457.0314329.58673729.0942985.8617704-3.06469922-.15890802-5.78835241-1.66547825-7.60988389-3.9574208-.3174714.56076194-.49978171 1.21167363-.49978171 1.90536824 0 1.31404706.65223085 2.47224203 1.64236444 3.15218497-.60350999-.0198635-1.17401554-.1925232-1.67222562-.47366811v.04583885c0 1.83355406 1.27302891 3.36609966 2.96411421 3.71294696-.31118484.0886217-.63651445.1329326-.97441718.1329326-.2357461 0-.47149219-.0229194-.69466516-.0672303.47149219 1.5065703 1.83253297 2.6036468 3.44975116 2.632678-1.2651707 1.0160946-2.85724264 1.6196394-4.5891906 1.6196394-.29861172 0-.59093688-.0152796-.88011875-.0504227 1.63450624 1.0726291 3.57548241 1.6990934 5.66104951 1.6990934 6.79263079 0 10.50641749-5.7711113 10.50641749-10.7751859l-.0094298-.48894775c.7229547-.53478659 1.3516109-1.20250585 1.8419628-1.96190282-.6632323.30100846-1.3751855.50422736-2.1217148.59590507z" fill-rule="evenodd"/></symbol><symbol id="icon-social-youtube" viewBox="0 0 24 24"><path d="m10.1415 14.3973208-.0005625-5.19318431 4.863375 2.60554491zm9.963-7.92753362c-.6845625-.73643756-1.4518125-.73990314-1.803375-.7826454-2.518875-.18714178-6.2971875-.18714178-6.2971875-.18714178-.007875 0-3.7861875 0-6.3050625.18714178-.352125.04274226-1.1188125.04620784-1.8039375.7826454-.5394375.56084773-.7149375 1.8344515-.7149375 1.8344515s-.18 1.49597903-.18 2.99138042v1.4024082c0 1.495979.18 2.9913804.18 2.9913804s.1755 1.2736038.7149375 1.8344515c.685125.7364376 1.5845625.7133337 1.9850625.7901542 1.44.1420891 6.12.1859866 6.12.1859866s3.78225-.005776 6.301125-.1929178c.3515625-.0433198 1.1188125-.0467854 1.803375-.783223.5394375-.5608477.7155-1.8344515.7155-1.8344515s.18-1.4954014.18-2.9913804v-1.4024082c0-1.49540139-.18-2.99138042-.18-2.99138042s-.1760625-1.27360377-.7155-1.8344515z" fill-rule="evenodd"/></symbol><symbol id="icon-subject-medicine" viewBox="0 0 18 18"><path d="m12.5 8h-6.5c-1.65685425 0-3 1.34314575-3 3v1c0 1.6568542 1.34314575 3 3 3h1v-2h-.5c-.82842712 0-1.5-.6715729-1.5-1.5s.67157288-1.5 1.5-1.5h1.5 2 1 2c1.6568542 0 3-1.34314575 3-3v-1c0-1.65685425-1.3431458-3-3-3h-2v2h1.5c.8284271 0 1.5.67157288 1.5 1.5s-.6715729 1.5-1.5 1.5zm-5.5-1v-1h-3.5c-1.38071187 0-2.5-1.11928813-2.5-2.5s1.11928813-2.5 2.5-2.5h1.02786405c.46573528 0 .92507448.10843528 1.34164078.31671843l1.13382424.56691212c.06026365-1.05041141.93116291-1.88363055 1.99667093-1.88363055 1.1045695 0 2 .8954305 2 2h2c2.209139 0 4 1.790861 4 4v1c0 2.209139-1.790861 4-4 4h-2v1h2c1.1045695 0 2 .8954305 2 2s-.8954305 2-2 2h-2c0 1.1045695-.8954305 2-2 2s-2-.8954305-2-2h-1c-2.209139 0-4-1.790861-4-4v-1c0-2.209139 1.790861-4 4-4zm0-2v-2.05652691c-.14564246-.03538148-.28733393-.08714006-.42229124-.15461871l-1.15541752-.57770876c-.27771087-.13885544-.583937-.21114562-.89442719-.21114562h-1.02786405c-.82842712 0-1.5.67157288-1.5 1.5s.67157288 1.5 1.5 1.5zm4 1v1h1.5c.2761424 0 .5-.22385763.5-.5s-.2238576-.5-.5-.5zm-1 1v-5c0-.55228475-.44771525-1-1-1s-1 .44771525-1 1v5zm-2 4v5c0 .5522847.44771525 1 1 1s1-.4477153 1-1v-5zm3 2v2h2c.5522847 0 1-.4477153 1-1s-.4477153-1-1-1zm-4-1v-1h-.5c-.27614237 0-.5.2238576-.5.5s.22385763.5.5.5zm-3.5-9h1c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5z" fill-rule="evenodd"/></symbol><symbol id="icon-success" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm3.4860198 4.98163161-4.71802968 5.50657859-2.62834168-2.02300024c-.42862421-.36730544-1.06564993-.30775346-1.42283677.13301307-.35718685.44076653-.29927542 1.0958383.12934879 1.46314377l3.40735508 2.7323063c.42215801.3385221 1.03700951.2798252 1.38749189-.1324571l5.38450527-6.33394549c.3613513-.43716226.3096573-1.09278382-.115462-1.46437175-.4251192-.37158792-1.0626796-.31842941-1.4240309.11873285z" fill-rule="evenodd"/></symbol><symbol id="icon-table" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587l-4.0059107-.001.001.001h-1l-.001-.001h-5l.001.001h-1l-.001-.001-3.00391071.001c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm-11.0059107 5h-3.999v6.9941413c0 .5572961.44630695 1.0058587.99508929 1.0058587h3.00391071zm6 0h-5v8h5zm5.0059107-4h-4.0059107v3h5.001v1h-5.001v7.999l4.0059107.001c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-12.5049107 9c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.2238576.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238576-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm-6-2c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.2238576.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.2238576-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.2238576.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.2238576-.5-.5s.22385763-.5.5-.5zm-6-2c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-1c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm12 0c.2761424 0 .5.22385763.5.5s-.2238576.5-.5.5h-2c-.2761424 0-.5-.22385763-.5-.5s.2238576-.5.5-.5zm-6 0c.27614237 0 .5.22385763.5.5s-.22385763.5-.5.5h-2c-.27614237 0-.5-.22385763-.5-.5s.22385763-.5.5-.5zm1.499-5h-5v3h5zm-6 0h-3.00391071c-.54871518 0-.99508929.44887827-.99508929 1.00585866v1.99414134h3.999z" fill-rule="evenodd"/></symbol><symbol id="icon-tick-circle" viewBox="0 0 24 24"><path d="m12 2c5.5228475 0 10 4.4771525 10 10s-4.4771525 10-10 10-10-4.4771525-10-10 4.4771525-10 10-10zm0 1c-4.97056275 0-9 4.02943725-9 9 0 4.9705627 4.02943725 9 9 9 4.9705627 0 9-4.0294373 9-9 0-4.97056275-4.0294373-9-9-9zm4.2199868 5.36606669c.3613514-.43716226.9989118-.49032077 1.424031-.11873285s.4768133 1.02720949.115462 1.46437175l-6.093335 6.94397871c-.3622945.4128716-.9897871.4562317-1.4054264.0971157l-3.89719065-3.3672071c-.42862421-.3673054-.48653564-1.0223772-.1293488-1.4631437s.99421256-.5003185 1.42283677-.1330131l3.11097438 2.6987741z" fill-rule="evenodd"/></symbol><symbol id="icon-tick" viewBox="0 0 16 16"><path d="m6.76799012 9.21106946-3.1109744-2.58349728c-.42862421-.35161617-1.06564993-.29460792-1.42283677.12733148s-.29927541 1.04903009.1293488 1.40064626l3.91576307 3.23873978c.41034319.3393961 1.01467563.2976897 1.37450571-.0948578l6.10568327-6.660841c.3613513-.41848908.3096572-1.04610608-.115462-1.4018218-.4251192-.35571573-1.0626796-.30482786-1.424031.11366122z" fill-rule="evenodd"/></symbol><symbol id="icon-update" viewBox="0 0 18 18"><path d="m1 13v1c0 .5522847.44771525 1 1 1h14c.5522847 0 1-.4477153 1-1v-1h-1v-10h-14v10zm16-1h1v2c0 1.1045695-.8954305 2-2 2h-14c-1.1045695 0-2-.8954305-2-2v-2h1v-9c0-.55228475.44771525-1 1-1h14c.5522847 0 1 .44771525 1 1zm-1 0v1h-4.5857864l-1 1h-2.82842716l-1-1h-4.58578644v-1h5l1 1h2l1-1zm-13-8h12v7h-12zm1 1v5h10v-5zm1 1h4v1h-4zm0 2h4v1h-4z" fill-rule="evenodd"/></symbol><symbol id="icon-upload" viewBox="0 0 18 18"><path d="m10.0046024 0c.5497429 0 1.3179837.32258606 1.707238.71184039l4.5763192 4.57631922c.3931386.39313859.7118404 1.16760135.7118404 1.71431368v8.98899651c0 1.1092806-.8945138 2.0085302-1.9940603 2.0085302h-12.01187942c-1.10128908 0-1.99406028-.8926228-1.99406028-1.9950893v-14.00982141c0-1.10185739.88743329-1.99508929 1.99961498-1.99508929zm0 1h-7.00498742c-.55709576 0-.99961498.44271433-.99961498.99508929v14.00982141c0 .5500396.44491393.9950893.99406028.9950893h12.01187942c.5463747 0 .9940603-.4506622.9940603-1.0085302v-8.98899651c0-.28393444-.2150684-.80332809-.4189472-1.0072069l-4.5763192-4.57631922c-.2038461-.20384606-.718603-.41894717-1.0001312-.41894717zm-1.85576936 4.14572769c.19483374-.19483375.51177826-.19377714.70556874.00001334l2.59099082 2.59099079c.1948411.19484112.1904373.51514474.0027906.70279143-.1932998.19329987-.5046517.19237083-.7001856-.00692852l-1.74638687-1.7800176v6.14827687c0 .2717771-.23193359.492096-.5.492096-.27614237 0-.5-.216372-.5-.492096v-6.14827641l-1.74627892 1.77990922c-.1933927.1971171-.51252214.19455839-.70016883.0069117-.19329987-.19329988-.19100584-.50899493.00277731-.70277808z" fill-rule="evenodd"/></symbol><symbol id="icon-video" viewBox="0 0 18 18"><path d="m16.0049107 2c1.1018574 0 1.9950893.89706013 1.9950893 2.00585866v9.98828264c0 1.1078052-.8926228 2.0058587-1.9950893 2.0058587h-14.00982141c-1.10185739 0-1.99508929-.8970601-1.99508929-2.0058587v-9.98828264c0-1.10780515.8926228-2.00585866 1.99508929-2.00585866zm0 1h-14.00982141c-.54871518 0-.99508929.44887827-.99508929 1.00585866v9.98828264c0 .5572961.44630695 1.0058587.99508929 1.0058587h14.00982141c.5487152 0 .9950893-.4488783.9950893-1.0058587v-9.98828264c0-.55729607-.446307-1.00585866-.9950893-1.00585866zm-8.30912922 2.24944486 4.60460462 2.73982242c.9365543.55726659.9290753 1.46522435 0 2.01804082l-4.60460462 2.7398224c-.93655425.5572666-1.69578148.1645632-1.69578148-.8937585v-5.71016863c0-1.05087579.76670616-1.446575 1.69578148-.89375851zm-.67492769.96085624v5.5750128c0 .2995102-.10753745.2442517.16578928.0847713l4.58452283-2.67497259c.3050619-.17799716.3051624-.21655446 0-.39461026l-4.58452283-2.67497264c-.26630747-.15538481-.16578928-.20699944-.16578928.08477139z" fill-rule="evenodd"/></symbol><symbol id="icon-warning" viewBox="0 0 18 18"><path d="m9 11.75c.69035594 0 1.25.5596441 1.25 1.25s-.55964406 1.25-1.25 1.25-1.25-.5596441-1.25-1.25.55964406-1.25 1.25-1.25zm.41320045-7.75c.55228475 0 1.00000005.44771525 1.00000005 1l-.0034543.08304548-.3333333 4c-.043191.51829212-.47645714.91695452-.99654578.91695452h-.15973424c-.52008864 0-.95335475-.3986624-.99654576-.91695452l-.33333333-4c-.04586475-.55037702.36312325-1.03372649.91350028-1.07959124l.04148683-.00259031zm-.41320045 14c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9z" fill-rule="evenodd"/></symbol><symbol id="icon-checklist-banner" viewBox="0 0 56.69 56.69"><path style="fill:none" d="M0 0h56.69v56.69H0z"/><clipPath id="b"><use xlink:href="#a" style="overflow:visible"/></clipPath><path d="M21.14 34.46c0-6.77 5.48-12.26 12.24-12.26s12.24 5.49 12.24 12.26-5.48 12.26-12.24 12.26c-6.76-.01-12.24-5.49-12.24-12.26zm19.33 10.66 10.23 9.22s1.21 1.09 2.3-.12l2.09-2.32s1.09-1.21-.12-2.3l-10.23-9.22m-19.29-5.92c0-4.38 3.55-7.94 7.93-7.94s7.93 3.55 7.93 7.94c0 4.38-3.55 7.94-7.93 7.94-4.38-.01-7.93-3.56-7.93-7.94zm17.58 12.99 4.14-4.81" style="clip-path:url(#b);fill:none;stroke:#01324b;stroke-width:2;stroke-linecap:round"/><path d="M8.26 9.75H28.6M8.26 15.98H28.6m-20.34 6.2h12.5m14.42-5.2V4.86s0-2.93-2.93-2.93H4.13s-2.93 0-2.93 2.93v37.57s0 2.93 2.93 2.93h15.01M8.26 9.75H28.6M8.26 15.98H28.6m-20.34 6.2h12.5" style="clip-path:url(#b);fill:none;stroke:#01324b;stroke-width:2;stroke-linecap:round;stroke-linejoin:round"/></symbol><symbol id="icon-chevron-down" viewBox="0 0 16 16"><path d="m5.58578644 3-3.29289322-3.29289322c-.39052429-.39052429-.39052429-1.02368927 0-1.41421356s1.02368927-.39052429 1.41421356 0l4 4c.39052429.39052429.39052429 1.02368927 0 1.41421356l-4 4c-.39052429.39052429-1.02368927.39052429-1.41421356 0s-.39052429-1.02368927 0-1.41421356z" fill-rule="evenodd" transform="matrix(0 1 -1 0 11 1)"/></symbol><symbol id="icon-eds-i-arrow-right-medium" viewBox="0 0 24 24"><path d="m12.728 3.293 7.98 7.99a.996.996 0 0 1 .281.561l.011.157c0 .32-.15.605-.384.788l-7.908 7.918a1 1 0 0 1-1.416-1.414L17.576 13H4a1 1 0 0 1 0-2h13.598l-6.285-6.293a1 1 0 0 1-.082-1.32l.083-.095a1 1 0 0 1 1.414.001Z"/></symbol><symbol id="icon-eds-i-chevron-down-medium" viewBox="0 0 16 16"><path d="m2.00087166 7h4.99912834v-4.99912834c0-.55276616.44386482-1.00087166 1-1.00087166.55228475 0 1 .44463086 1 1.00087166v4.99912834h4.9991283c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-4.9991283v4.9991283c0 .5527662-.44386482 1.0008717-1 1.0008717-.55228475 0-1-.4446309-1-1.0008717v-4.9991283h-4.99912834c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-eds-i-chevron-down-small" viewBox="0 0 16 16"><path d="M13.692 5.278a1 1 0 0 1 .03 1.414L9.103 11.51a1.491 1.491 0 0 1-2.188.019L2.278 6.692a1 1 0 0 1 1.444-1.384L8 9.771l4.278-4.463a1 1 0 0 1 1.318-.111l.096.081Z"/></symbol><symbol id="icon-eds-i-chevron-right-medium" viewBox="0 0 10 10"><path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/></symbol><symbol id="icon-eds-i-chevron-right-small" viewBox="0 0 10 10"><path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/></symbol><symbol id="icon-eds-i-chevron-up-medium" viewBox="0 0 16 16"><path d="m2.00087166 7h11.99825664c.5527662 0 1.0008717.44386482 1.0008717 1 0 .55228475-.4446309 1-1.0008717 1h-11.99825664c-.55276616 0-1.00087166-.44386482-1.00087166-1 0-.55228475.44463086-1 1.00087166-1z" fill-rule="evenodd"/></symbol><symbol id="icon-eds-i-close-medium" viewBox="0 0 16 16"><path d="m2.29679575 12.2772478c-.39658757.3965876-.39438847 1.0328109-.00062148 1.4265779.39651227.3965123 1.03246768.3934888 1.42657791-.0006214l4.27724782-4.27724787 4.2772478 4.27724787c.3965876.3965875 1.0328109.3943884 1.4265779.0006214.3965123-.3965122.3934888-1.0324677-.0006214-1.4265779l-4.27724787-4.2772478 4.27724787-4.27724782c.3965875-.39658757.3943884-1.03281091.0006214-1.42657791-.3965122-.39651226-1.0324677-.39348875-1.4265779.00062148l-4.2772478 4.27724782-4.27724782-4.27724782c-.39658757-.39658757-1.03281091-.39438847-1.42657791-.00062148-.39651226.39651227-.39348875 1.03246768.00062148 1.42657791l4.27724782 4.27724782z" fill-rule="evenodd"/></symbol><symbol id="icon-eds-i-download-medium" viewBox="0 0 16 16"><path d="m12.9975267 12.999368c.5467123 0 1.0024733.4478567 1.0024733 1.000316 0 .5563109-.4488226 1.000316-1.0024733 1.000316h-9.99505341c-.54671233 0-1.00247329-.4478567-1.00247329-1.000316 0-.5563109.44882258-1.000316 1.00247329-1.000316zm-4.9975267-11.999368c.55228475 0 1 .44497754 1 .99589209v6.80214418l2.4816273-2.48241149c.3928222-.39294628 1.0219732-.4006883 1.4030652-.01947579.3911302.39125371.3914806 1.02525073-.0001404 1.41699553l-4.17620792 4.17752758c-.39120769.3913313-1.02508144.3917306-1.41671995-.0000316l-4.17639421-4.17771394c-.39122513-.39134876-.39767006-1.01940351-.01657797-1.40061601.39113012-.39125372 1.02337105-.3931606 1.41951349.00310701l2.48183446 2.48261871v-6.80214418c0-.55001601.44386482-.99589209 1-.99589209z" fill-rule="evenodd"/></symbol><symbol id="icon-eds-i-info-filled-medium" viewBox="0 0 18 18"><path d="m9 0c4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9zm0 7h-1.5l-.11662113.00672773c-.49733868.05776511-.88337887.48043643-.88337887.99327227 0 .47338693.32893365.86994729.77070917.97358929l.1126697.01968298.11662113.00672773h.5v3h-.5l-.11662113.0067277c-.42082504.0488782-.76196299.3590206-.85696816.7639815l-.01968298.1126697-.00672773.1166211.00672773.1166211c.04887817.4208251.35902055.761963.76398144.8569682l.1126697.019683.11662113.0067277h3l.1166211-.0067277c.4973387-.0577651.8833789-.4804365.8833789-.9932723 0-.4733869-.3289337-.8699473-.7707092-.9735893l-.1126697-.019683-.1166211-.0067277h-.5v-4l-.00672773-.11662113c-.04887817-.42082504-.35902055-.76196299-.76398144-.85696816l-.1126697-.01968298zm0-3.25c-.69035594 0-1.25.55964406-1.25 1.25s.55964406 1.25 1.25 1.25 1.25-.55964406 1.25-1.25-.55964406-1.25-1.25-1.25z" fill-rule="evenodd"/></symbol><symbol id="icon-eds-i-mail-medium" viewBox="0 0 24 24"><path d="m19.462 0c1.413 0 2.538 1.184 2.538 2.619v12.762c0 1.435-1.125 2.619-2.538 2.619h-16.924c-1.413 0-2.538-1.184-2.538-2.619v-12.762c0-1.435 1.125-2.619 2.538-2.619zm.538 5.158-7.378 6.258a2.549 2.549 0 0 1 -3.253-.008l-7.369-6.248v10.222c0 .353.253.619.538.619h16.924c.285 0 .538-.266.538-.619zm-.538-3.158h-16.924c-.264 0-.5.228-.534.542l8.65 7.334c.2.165.492.165.684.007l8.656-7.342-.001-.025c-.044-.3-.274-.516-.531-.516z"/></symbol><symbol id="icon-eds-i-menu-medium" viewBox="0 0 24 24"><path d="M21 4a1 1 0 0 1 0 2H3a1 1 0 1 1 0-2h18Zm-4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h14Zm4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h18Z"/></symbol><symbol id="icon-eds-i-search-medium" viewBox="0 0 24 24"><path d="M11 1c5.523 0 10 4.477 10 10 0 2.4-.846 4.604-2.256 6.328l3.963 3.965a1 1 0 0 1-1.414 1.414l-3.965-3.963A9.959 9.959 0 0 1 11 21C5.477 21 1 16.523 1 11S5.477 1 11 1Zm0 2a8 8 0 1 0 0 16 8 8 0 0 0 0-16Z"/></symbol><symbol id="icon-eds-i-user-single-medium" viewBox="0 0 24 24"><path d="M12 1a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm-.406 9.008a8.965 8.965 0 0 1 6.596 2.494A9.161 9.161 0 0 1 21 21.025V22a1 1 0 0 1-1 1H4a1 1 0 0 1-1-1v-.985c.05-4.825 3.815-8.777 8.594-9.007Zm.39 1.992-.299.006c-3.63.175-6.518 3.127-6.678 6.775L5 21h13.998l-.009-.268a7.157 7.157 0 0 0-1.97-4.573l-.214-.213A6.967 6.967 0 0 0 11.984 14Z"/></symbol><symbol id="icon-eds-i-warning-filled-medium" viewBox="0 0 18 18"><path d="m9 11.75c.69035594 0 1.25.5596441 1.25 1.25s-.55964406 1.25-1.25 1.25-1.25-.5596441-1.25-1.25.55964406-1.25 1.25-1.25zm.41320045-7.75c.55228475 0 1.00000005.44771525 1.00000005 1l-.0034543.08304548-.3333333 4c-.043191.51829212-.47645714.91695452-.99654578.91695452h-.15973424c-.52008864 0-.95335475-.3986624-.99654576-.91695452l-.33333333-4c-.04586475-.55037702.36312325-1.03372649.91350028-1.07959124l.04148683-.00259031zm-.41320045 14c-4.97056275 0-9-4.0294373-9-9 0-4.97056275 4.02943725-9 9-9 4.9705627 0 9 4.02943725 9 9 0 4.9705627-4.0294373 9-9 9z" fill-rule="evenodd"/></symbol><symbol id="icon-expand-image" viewBox="0 0 18 18"><path d="m7.49754099 11.9178212c.38955542-.3895554.38761957-1.0207846-.00290473-1.4113089-.39324695-.3932469-1.02238878-.3918247-1.41130883-.0029047l-4.10273549 4.1027355.00055454-3.5103985c.00008852-.5603185-.44832171-1.006032-1.00155062-1.0059446-.53903074.0000852-.97857527.4487442-.97866268 1.0021075l-.00093318 5.9072465c-.00008751.553948.44841131 1.001882 1.00174994 1.0017946l5.906983-.0009331c.5539233-.0000875 1.00197907-.4486389 1.00206646-1.0018679.00008515-.5390307-.45026621-.9784332-1.00588841-.9783454l-3.51010549.0005545zm3.00571741-5.83449376c-.3895554.38955541-.3876196 1.02078454.0029047 1.41130883.393247.39324696 1.0223888.39182478 1.4113089.00290473l4.1027355-4.10273549-.0005546 3.5103985c-.0000885.56031852.4483217 1.006032 1.0015506 1.00594461.5390308-.00008516.9785753-.44874418.9786627-1.00210749l.0009332-5.9072465c.0000875-.553948-.4484113-1.00188204-1.0017499-1.00179463l-5.906983.00093313c-.5539233.00008751-1.0019791.44863892-1.0020665 1.00186784-.0000852.53903074.4502662.97843325 1.0058884.97834547l3.5101055-.00055449z" fill-rule="evenodd"/></symbol><symbol id="icon-github" viewBox="0 0 100 100"><path fill-rule="evenodd" clip-rule="evenodd" d="M48.854 0C21.839 0 0 22 0 49.217c0 21.756 13.993 40.172 33.405 46.69 2.427.49 3.316-1.059 3.316-2.362 0-1.141-.08-5.052-.08-9.127-13.59 2.934-16.42-5.867-16.42-5.867-2.184-5.704-5.42-7.17-5.42-7.17-4.448-3.015.324-3.015.324-3.015 4.934.326 7.523 5.052 7.523 5.052 4.367 7.496 11.404 5.378 14.235 4.074.404-3.178 1.699-5.378 3.074-6.6-10.839-1.141-22.243-5.378-22.243-24.283 0-5.378 1.94-9.778 5.014-13.2-.485-1.222-2.184-6.275.486-13.038 0 0 4.125-1.304 13.426 5.052a46.97 46.97 0 0 1 12.214-1.63c4.125 0 8.33.571 12.213 1.63 9.302-6.356 13.427-5.052 13.427-5.052 2.67 6.763.97 11.816.485 13.038 3.155 3.422 5.015 7.822 5.015 13.2 0 18.905-11.404 23.06-22.324 24.283 1.78 1.548 3.316 4.481 3.316 9.126 0 6.6-.08 11.897-.08 13.526 0 1.304.89 2.853 3.316 2.364 19.412-6.52 33.405-24.935 33.405-46.691C97.707 22 75.788 0 48.854 0z"/></symbol><symbol id="icon-springer-arrow-left"><path d="M15 7a1 1 0 000-2H3.385l2.482-2.482a.994.994 0 00.02-1.403 1.001 1.001 0 00-1.417 0L.294 5.292a1.001 1.001 0 000 1.416l4.176 4.177a.991.991 0 001.4.016 1 1 0 00-.003-1.42L3.385 7H15z"/></symbol><symbol id="icon-springer-arrow-right"><path d="M1 7a1 1 0 010-2h11.615l-2.482-2.482a.994.994 0 01-.02-1.403 1.001 1.001 0 011.417 0l4.176 4.177a1.001 1.001 0 010 1.416l-4.176 4.177a.991.991 0 01-1.4.016 1 1 0 01.003-1.42L12.615 7H1z"/></symbol><symbol id="icon-submit-open" viewBox="0 0 16 17"><path d="M12 0c1.10457 0 2 .895431 2 2v5c0 .276142-.223858.5-.5.5S13 7.276142 13 7V2c0-.512836-.38604-.935507-.883379-.993272L12 1H6v3c0 1.10457-.89543 2-2 2H1v8c0 .512836.38604.935507.883379.993272L2 15h6.5c.276142 0 .5.223858.5.5s-.223858.5-.5.5H2c-1.104569 0-2-.89543-2-2V5.828427c0-.530433.210714-1.039141.585786-1.414213L4.414214.585786C4.789286.210714 5.297994 0 5.828427 0H12Zm3.41 11.14c.250899.250899.250274.659726 0 .91-.242954.242954-.649606.245216-.9-.01l-1.863671-1.900337.001043 5.869492c0 .356992-.289839.637138-.647372.637138-.347077 0-.647371-.285256-.647371-.637138l-.001043-5.869492L9.5 12.04c-.253166.258042-.649726.260274-.9.01-.242954-.242954-.252269-.657731 0-.91l2.942184-2.951303c.250908-.250909.66127-.252277.91353-.000017L15.41 11.14ZM5 1.413 1.413 5H4c.552285 0 1-.447715 1-1V1.413ZM11 3c.276142 0 .5.223858.5.5s-.223858.5-.5.5H7.5c-.276142 0-.5-.223858-.5-.5s.223858-.5.5-.5H11Zm0 2c.276142 0 .5.223858.5.5s-.223858.5-.5.5H7.5c-.276142 0-.5-.223858-.5-.5s.223858-.5.5-.5H11Z" fill-rule="nonzero"/></symbol></svg> </div> </body> </html>