Document Zbl 1545.65027

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Appl. Probab.</a> <a href="/?q=in%3A516257" title="Articles in this Issue">34, No. 2, 2326-2363 (2024)</a>. </div> <div class="abstract">Summary: We propose an explicit drift-randomised Milstein scheme for both McKean-Vlasov stochastic differential equations and associated high-dimensional interacting particle systems with common noise. By using a drift randomisation step in space and measure, we establish the scheme’s strong convergence rate of 1 under reduced regularity assumptions on the drift coefficient: no classical (Euclidean) derivatives in space or measure derivatives (e.g., Lions/Fréchet) are required. The main result is established by enriching the concepts of bistability and consistency of numerical schemes used previously for standard SDE. We introduce certain Spijker-type norms (and associated Banach spaces) to deal with the interaction of particles present in the stochastic systems being analysed. A discussion of the scheme’s complexity is provided.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7899447">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C30" title="MSC2020">65C30</a> </td> <td class="space"> Numerical solutions to stochastic differential and integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H10" title="MSC2020">60H10</a> </td> <td class="space"> Stochastic ordinary differential equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H35" title="MSC2020">60H35</a> </td> <td class="space"> Computational methods for stochastic equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K35" title="MSC2020">60K35</a> </td> <td class="space"> Interacting random processes; statistical mechanics type models; percolation theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C05" title="MSC2020">65C05</a> </td> <td class="space"> Monte Carlo methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C35" title="MSC2020">65C35</a> </td> <td class="space"> Stochastic particle methods </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Abistability">bistability</a>; <a href="/?q=ut%3Acommon+noise">common noise</a>; <a href="/?q=ut%3Adrift+randomisation">drift randomisation</a>; <a href="/?q=ut%3AMcKean-Vlasov+equations">McKean-Vlasov equations</a>; <a href="/?q=ut%3AMilstein+scheme">Milstein scheme</a>; <a href="/?q=ut%3Anondifferentiable+drift">nondifferentiable drift</a>; <a href="/?q=ut%3Astochastic+interacting+particle+systems">stochastic interacting particle systems</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 1545.65027" data-ciurl="/ci/07899447" data-biburl="/bibtex/07899447.bib" data-amsurl="/amsrefs/07899447.bib" data-xmlurl="/xml/07899447.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07899447.pdf" title="Zbl 1545.65027 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/10.1214/23-AAP2024" aria-label="DOI for “An explicit Milstein-type scheme for interacting particle systems and McKean-Vlasov SDEs with common noise and non-differentiable drift coefficients”" title="10.1214/23-AAP2024">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2208.10052"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://projecteuclid.org/journals/annals-of-applied-probability/volume-34/issue-2/An-explicit-Milstein-type-scheme-for-interacting-particle-systems-and/10.1214/23-AAP2024.full" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">ADAMS, D., DOS REIS, G., RAVAILLE, R., SALKELD, W. and TUGAUT, J. 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