{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T22:00:53Z","timestamp":1775599253519,"version":"3.50.1"},"reference-count":41,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T00:00:00Z","timestamp":1611792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,1,21]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present two fully probabilistic Euler schemes, one explicit and one implicit, for the simulation of McKean\u2013Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth and random initial condition. We provide a pathwise propagation of chaos result and show strong convergence for both schemes on the consequent particle system. The explicit scheme attains the standard $1\/2$ rate in stepsize. From a technical point of view, we successfully use stopping times to prove the convergence of the implicit method; although we avoid them altogether for the explicit one. The combination of particle interactions and random initial condition makes the proofs technically more involved. Numerical tests recover the theoretical convergence rates and illustrate a computational complexity advantage of the explicit over the implicit scheme. Comparative analysis is carried out on a stylized non-Lipschitz MV-SDE and a mean-field model for FitzHugh\u2013Nagumo neurons. We provide numerical tests illustrating particle corruption effect where one single particle diverging can \u2018corrupt\u2019 the whole particle system. Moreover, the more particles in the system the more likely this divergence is to occur.<\/jats:p>","DOI":"10.1093\/imanum\/draa099","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T20:21:30Z","timestamp":1607977290000},"page":"874-922","source":"Crossref","is-referenced-by-count":53,"title":["Simulation of McKean\u2013Vlasov SDEs with super-linear growth"],"prefix":"10.1093","volume":"42","author":[{"given":"Gon\u00e7alo","family":"dos Reis","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK and Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA), FCT, UNL, Quinta da Torre, 2829 -516 Caparica, PT"}]},{"given":"Stefan","family":"Engelhardt","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik, Friedrich-Schiller-Universit\u00e4t Jena, Ernst-Abbe-Platz 2, 07743 Jena, DE"}]},{"given":"Greig","family":"Smith","sequence":"additional","affiliation":[{"name":"University of Edinburgh, Maxwell Institute for Mathematical Sciences, James Clerk Maxwell Building, Mayfield Rd, Edinburgh EH9 3FD, UK"}]}],"member":"286","published-online":{"date-parts":[[2021,1,28]]},"reference":[{"key":"2022011721233372000_ref1","article-title":"Large deviations and exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts","author":"Adams","year":"2020"},{"key":"2022011721233372000_ref2","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/2190-8567-2-10","article-title":"Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons","volume":"2","author":"Baladron","year":"2012","journal-title":"J. Math. Neurosci."},{"key":"2022011721233372000_ref3","doi-asserted-by":"crossref","first-page":"2179","DOI":"10.1142\/S0218202511005702","article-title":"Stochastic mean-field limit: non-Lipschitz forces and swarming","volume":"21","author":"Bolley","year":"2011","journal-title":"Math. Models Methods Appl. Sci."},{"key":"2022011721233372000_ref4","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1186\/s13408-015-0031-8","article-title":"Clarification and complement to \u2018mean-field description and propagation of chaos in networks of Hodgkin\u2013Huxley and FitzHugh\u2013Nagumo neurons\u2019","volume":"5","author":"Bossy","year":"2015","journal-title":"J. Math. Neurosci."},{"key":"2022011721233372000_ref5","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1090\/S0025-5718-97-00776-X","article-title":"A stochastic particle method for the McKean-Vlasov and the Burgers equation","volume":"66","author":"Bossy","year":"1997","journal-title":"Math. Comput. Amer. Math. Soc."},{"key":"2022011721233372000_ref6","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1214\/17-EJP25","article-title":"Uniform in time interacting particle approximations for nonlinear equations of Patlak-Keller-Segel type","volume":"22","author":"Budhiraja","year":"2017","journal-title":"Electron. J. Probab"},{"key":"2022011721233372000_ref7","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611974249","article-title":"Society for Industrial and Applied Mathematics (SIAM)","volume-title":"Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications","author":"Carmona","year":"2016"},{"key":"2022011721233372000_ref8","volume-title":"Probabilistic Theory of Mean Field Games with Applications: I: Mean field FBSDEs, control, and games","author":"Carmona","year":"2018"},{"key":"2022011721233372000_ref9","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1137\/15M1017788","article-title":"An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients","volume":"7","author":"Chassagneux","year":"2016","journal-title":"SIAM J. Financial Math."},{"key":"2022011721233372000_ref10","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1214\/19-EJP298","article-title":"From the master equation to mean field game limit theory: a central limit theorem","volume":"24","author":"Delarue","year":"2019","journal-title":"Electron. J. Probab."},{"key":"2022011721233372000_ref11","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1214\/18-AAP1416","article-title":"Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the functional iterated logarithm law","volume":"29","author":"dos Reis","year":"2019","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref12","article-title":"Importance sampling for McKean-Vlasov SDEs","author":"dos Reis","year":"2018"},{"key":"2022011721233372000_ref13","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1017\/S0956792511000052","article-title":"Phase transition in a rechargeable lithium battery","volume":"22","author":"Dreyer","year":"2011","journal-title":"Eur. J. Appl. Math."},{"key":"2022011721233372000_ref14","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1214\/19-AAP1507","article-title":"Adaptive Euler-Maruyama method for SDEs with nonglobally Lipschitz drift","volume":"30","author":"Fang","year":"2020","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref15","first-page":"2","article-title":"Erratum: Concentration bounds for stochastic approximations [mr2988393]","volume":"17","author":"Frikha","year":"2012","journal-title":"Electron. Commun. Probab."},{"key":"2022011721233372000_ref16","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2018.05.059","article-title":"Analytical approximations of non-linear SDEs of McKean-Vlasov type","author":"Gobet","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"2022011721233372000_ref17","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1137\/19M1258116","article-title":"Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods","volume":"18","author":"Gomes","year":"2020","journal-title":"Multiscale Model. Simul."},{"key":"2022011721233372000_ref18","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1007\/s00161-018-0629-7","article-title":"Stochastic many-particle model for LFP electrodes","volume":"30","author":"Guhlke","year":"2018","journal-title":"Contin. Mech. Thermodyn."},{"key":"2022011721233372000_ref19","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1137\/S0036142901389530","article-title":"Strong convergence of Euler-type methods for nonlinear stochastic differential equations","volume":"40","author":"Higham","year":"2002","journal-title":"SIAM J. Numer. Anal."},{"key":"2022011721233372000_ref20","first-page":"1563","article-title":"Strong and weak divergence in finite time of Euler\u2019s method for stochastic differential equations with non-globally Lipschitz continuous coefficients","volume":"467","author":"Hutzenthaler","year":"2011","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"2022011721233372000_ref21","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1214\/11-AAP803","article-title":"Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients","volume":"22","author":"Hutzenthaler","year":"2012","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref22","article-title":"Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems","author":"Jourdain","year":"2020"},{"key":"2022011721233372000_ref23","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-93839-4","volume-title":"Theoretical Statistics","author":"Keener","year":"2010"},{"key":"2022011721233372000_ref24","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1515\/mcma.1997.3.4.327","article-title":"Weak rate of convergence for an Euler scheme of nonlinear SDE\u2019s","volume":"3","author":"Kohatsu-Higa","year":"1997","journal-title":"Monte Carlo Methods Appl."},{"key":"2022011721233372000_ref25","doi-asserted-by":"crossref","DOI":"10.1214\/18-ECP150","article-title":"On a strong form of propagation of chaos for McKean-Vlasov equations","volume":"23","author":"Lacker","year":"2018","journal-title":"Electron. Commun. Probab."},{"key":"2022011721233372000_ref26","doi-asserted-by":"crossref","first-page":"2563","DOI":"10.1214\/14-AAP1056","article-title":"Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs","volume":"25","author":"Lionnet","year":"2015","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref27","doi-asserted-by":"crossref","first-page":"2544","DOI":"10.1214\/17-AAP1366","article-title":"Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth","volume":"28","author":"Lionnet","year":"2018","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref28","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1214\/aoap\/1050689593","article-title":"Convergence to equilibrium for granular media equations and their Euler schemes","volume":"13","author":"Malrieu","year":"2003","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref29","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/3-540-31186-6_21","article-title":"Concentration inequalities for Euler schemes","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods, 2004","author":"Malrieu","year":"2006"},{"key":"2022011721233372000_ref30","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.cam.2012.08.015","article-title":"Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients","volume":"238","author":"Mao","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"2022011721233372000_ref31","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1214\/19-AAP1499","article-title":"Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions","volume":"30","author":"Mehri","year":"2020","journal-title":"Ann. Appl. Probab."},{"key":"2022011721233372000_ref32","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/BFb0093177","article-title":"Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models","volume-title":"Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995)","author":"M\u00e9l\u00e9ard","year":"1996"},{"key":"2022011721233372000_ref33","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1137\/040612026","article-title":"Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients","volume":"43","author":"Milstein","year":"2005","journal-title":"SIAM J. Numer. Anal."},{"key":"2022011721233372000_ref34","volume-title":"Inequalities Involving Functions and Their Integrals and Derivatives","author":"Mitrinovic","year":"2012"},{"key":"2022011721233372000_ref35","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-10061-5","article-title":"Volume 21 of Stochastic Modelling and Applied Probability","volume-title":"Stochastic Integration and Differential Equations","author":"Protter","year":"2005"},{"key":"2022011721233372000_ref36","article-title":"Mass transportation problems","volume-title":"Probability and its Applications (New York)","author":"Rachev","year":"1998"},{"key":"2022011721233372000_ref37","first-page":"10","article-title":"A note on tamed Euler approximations","volume-title":"Electron. Commun. Probab.","author":"Sabanis","year":"2013"},{"key":"2022011721233372000_ref38","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/BFb0085169","article-title":"Topics in propagation of chaos","volume":"1989","author":"Sznitman","year":"1991","journal-title":"Ecole d\u2019Et\u00e9 de Probabilit\u00e9s de Saint-Flour XIX \u2014"},{"key":"2022011721233372000_ref39","doi-asserted-by":"crossref","first-page":"1962","DOI":"10.1016\/j.physa.2013.01.042","article-title":"A stochastic Ginzburg-Landau equation with impulsive effects","volume":"392","author":"Tien","year":"2013","journal-title":"Physica A Stat. Mech. Appl."},{"key":"2022011721233372000_ref40","article-title":"Volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","volume-title":"Optimal Transport","author":"Villani","year":"2009"},{"key":"2022011721233372000_ref41","volume-title":"Nonlinear functional analysis and its applications: II\/B","author":"Zeidler","year":"1990"}],"container-title":["IMA Journal of Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imajna\/article-pdf\/42\/1\/874\/42098165\/draa099.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imajna\/article-pdf\/42\/1\/874\/42098165\/draa099.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,17]],"date-time":"2022-01-17T21:24:55Z","timestamp":1642454695000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imajna\/article\/42\/1\/874\/6121618"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,28]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1,28]]},"published-print":{"date-parts":[[2022,1,21]]}},"URL":"https:\/\/doi.org\/10.1093\/imanum\/draa099","relation":{},"ISSN":["0272-4979","1464-3642"],"issn-type":[{"value":"0272-4979","type":"print"},{"value":"1464-3642","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,1]]},"published":{"date-parts":[[2021,1,28]]}}}