dc.contributor.advisor Geiss, Christel dc.contributor.author Koskela, Emilia dc.date.accessioned 2023-06-13T04:52:33Z dc.date.available 2023-06-13T04:52:33Z dc.date.issued 2023 dc.identifier.uri https://jyx.jyu.fi/handle/123456789/87653 dc.description.abstract In this thesis we study stochastic McKean-Vlasov equations. These are stochastic differential equations where the coefficients depend also on the distribution of the solution. This dependency adds to the complexity of the equation so in this thesis we will study these equations using a discrete approximation. We focus on considering the existence of a unique strong solution to stochastic McKean-Vlasov equations using a discrete and recursive Euler-Maruyama approximation, as well as the convergence rate of the approximation. Our main source is the article Euler-Maruyama Approximations for Stochastic McKean-Vlasov Equations with Non-Lipschitz Coefficients written by Xiaojie Ding and Huijie Qiao, which we follow throughout this thesis. In the thesis we recall some preliminary theory surrounding stochastic processes and stochastic differential equations and introduce some results. We give the definitions for weak and strong solutions for the McKean-Vlasov equation as well as the definition for the martingale problem. We also introduce some useful inequalities. We give the assumptions under which we work in this thesis, such as the assumption that the coefficients of the McKean-Vlasov equations satisfy some non-Lipschitz conditions. One of the main results in this thesis is to show the existence of unique strong solutions. We approach this in two steps: first, we show the recursive construction of the Euler-Maruyama approximation. With this approximation we show that there exists a solution to the martingale problem and hence we get the existence of a weak solution. Then, using Ito’s formula we prove that pathwise uniqueness holds under our assumptions. After these two steps we show that the existence of a strong unique solution can be proven. We also investigate with the help of Ito’s formula the convergence rate of the Euler-Maruyama approximation used to show the existence of the solution. en dc.format.extent 45 dc.language.iso en dc.rights In Copyright dc.title Approximations for Stochastic McKean-Vlasov Equations with Non-Lipschitz Coefficients by an Euler-Maruyama Scheme dc.identifier.urn URN:NBN:fi:jyu-202306133721 dc.type.ontasot Master’s thesis en dc.type.ontasot Pro gradu -tutkielma fi dc.contributor.tiedekunta Matemaattis-luonnontieteellinen tiedekunta fi dc.contributor.tiedekunta Faculty of Sciences en dc.contributor.laitos Matematiikan ja tilastotieteen laitos fi dc.contributor.laitos Department of Mathematics and Statistics en dc.contributor.yliopisto Jyväskylän yliopisto fi dc.contributor.yliopisto University of Jyväskylä en dc.contributor.oppiaine Stokastiikka ja todennäköisyysteoria fi dc.contributor.oppiaine Stochastics and Probability en dc.rights.copyright © The Author(s) dc.rights.accesslevel openAccess dc.contributor.oppiainekoodi 4041 dc.subject.yso stokastiset prosessit dc.subject.yso matematiikka dc.subject.yso differentiaaliyhtälöt dc.subject.yso approksimointi dc.subject.yso stochastic processes dc.subject.yso mathematics dc.subject.yso differential equations dc.subject.yso approximation dc.rights.url https://rightsstatements.org/page/InC/1.0/
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