Approximations for Stochastic McKean-Vlasov Equations with Non-Lipschitz Coefficients by an Euler-Maruyama Scheme

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.