Document Zbl 1058.60095

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The claims less premiums cash flow is also Lévy with no common jumps with the returns. Then the reserve process of the insurance company itself belongs to the Lévy class. Special cases would include combinations of Brownian and Poisson processes for claims and returns as well as return processes with infinitely many jumps in every finite period in conjunction with Brownian claims. It is shown that the probability of ruin as a function of initial reserve level has a power function lower bound, provided that returns can get negative and losses are big enough. If the discretized compound return process satisfies an analogue of the Cramér condition, an upper ruin probability bound in a power function form exists as well. A more strong asymptotic property for the ruin probability in a power function form is obtained when discretized compound returns have a non-lattice distribution.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A9188">Alexis Derviz (Praha)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A2138954">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A2138954">Cited in <strong>79</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K99" title="MSC2020">60K99</a> </td> <td class="space"> Special processes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G70" title="MSC2020">60G70</a> </td> <td class="space"> Extreme value theory; extremal stochastic processes </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aruin+probability">ruin probability</a>; <a href="/?q=ut%3AL%C3%A9vy+process">Lévy process</a>; <a href="/?q=ut%3Areserves">reserves</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1058.60095" data-ciurl="/ci/02138954" data-biburl="/bibtex/02138954.bib" data-amsurl="/amsrefs/02138954.bib" data-xmlurl="/xml/02138954.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02138954.pdf" title="Zbl 1058.60095 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/S0304-4149(01)00148-X" aria-label="DOI for “Power tailed ruin probabilities in the presence of risky investments.”" title="10.1016/S0304-4149(01)00148-X">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bertoin, J., Lévy Processes (1996), Cambridge University Press: Cambridge University Press Cambridge · <a href="/0861.60003" class="nowrap">Zbl 0861.60003</a></td> </tr><tr> <td>[2]</td> <td class="space">Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events (1997), Springer: Springer Berlin · <a href="/0873.62116" class="nowrap">Zbl 0873.62116</a></td> </tr><tr> <td>[3]</td> <td class="space">Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Econom., 1, 55-72 (1982) · <a href="/0518.62083" class="nowrap">Zbl 0518.62083</a></td> </tr><tr> <td>[4]</td> <td class="space">Feller, W., An Introduction to Probability Theory and its Applications, Vol. II (1971), Wiley: Wiley New York · <a href="/0219.60003" class="nowrap">Zbl 0219.60003</a></td> </tr><tr> <td>[5]</td> <td class="space">Goldie, C. M., Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., 1, 126-166 (1991) · <a href="/0724.60076" class="nowrap">Zbl 0724.60076</a></td> </tr><tr> <td>[6]</td> <td class="space">Grandell, J., Aspects of Risk Theory (1991), Springer: Springer Berlin · <a href="/0717.62100" class="nowrap">Zbl 0717.62100</a></td> </tr><tr> <td>[7]</td> <td class="space">Grandell, J., 1998. Simple approximations of ruin probabilities. Preprint of Department of Mathematics, Royal Institute of Technology, Trita-Mat-1998-01.; Grandell, J., 1998. Simple approximations of ruin probabilities. Preprint of Department of Mathematics, Royal Institute of Technology, Trita-Mat-1998-01. · <a href="/0986.62086" class="nowrap">Zbl 0986.62086</a></td> </tr><tr> <td>[8]</td> <td class="space">Harrison, J. M., Ruin problems with compounding assets, J. Stochastic Proc. Appl., 5, 67-79 (1977) · <a href="/0361.60053" class="nowrap">Zbl 0361.60053</a></td> </tr><tr> <td>[9]</td> <td class="space">Kalashnikov, V., Two-sided bounds of ruin probabilities, Scand. Actuar. J., 1996, 1-18 (1996) · <a href="/0845.62075" class="nowrap">Zbl 0845.62075</a></td> </tr><tr> <td>[10]</td> <td class="space">Kalashnikov, V., Geometric Sums: Bounds for Rare Events with Applications (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · <a href="/0881.60043" class="nowrap">Zbl 0881.60043</a></td> </tr><tr> <td>[11]</td> <td class="space">Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta Math., 131, 207-248 (1973) · <a href="/0291.60029" class="nowrap">Zbl 0291.60029</a></td> </tr><tr> <td>[12]</td> <td class="space">Norberg, R., Ruin problems with assets and liabilities of diffusion type, J. Stochastic Proc. Appl., 81, 255-269 (1999) · <a href="/0962.60075" class="nowrap">Zbl 0962.60075</a></td> </tr><tr> <td>[13]</td> <td class="space">Nyrhinen, H., On the ruin probabilities in a general economic environment, J. Stochastic Proc. Appl., 83, 319-330 (1999) · <a href="/0997.60041" class="nowrap">Zbl 0997.60041</a></td> </tr><tr> <td>[14]</td> <td class="space">Paulsen, J., Risk theory in a stochastic environment, J. Stochastic Proc. Appl., 21, 327-361 (1993) · <a href="/0777.62098" class="nowrap">Zbl 0777.62098</a></td> </tr><tr> <td>[15]</td> <td class="space">Paulsen, J., Ruin theory with compounding assets—a survey, Insurance Math. Econom., 22, 3-16 (1998) · <a href="/0909.90115" class="nowrap">Zbl 0909.90115</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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