WELL-POSEDNESS AND EXPLICIT NUMERICAL SCHEMES FOR MCKEAN--VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

First, we consider McKean{Vlasov stochastic di erential equation (MV-SDE) and related highdimensional interacting particle systems with common noise and propose an explicit drift-randomized Milstein scheme. Under reduced regularity assumptions on the drift coe cient, we establish the strong convergence rate of 1 for the scheme by using a drift-randomization step in both space and measure. This reduces the need for derivatives in both space and measure components where the notion of Lions (Fr echet) derivative is used to di erentiate measure dependent function. Enhancing the concepts of bistability and consistency of the numerical schemes previously used for standard SDE allows one to establish the main result. To deal with the particle interaction in the stochastic systems, we introduce speci c Banach spaces and Spijker-type norms. Then, we consider MV{SDE driven by L evy noise and prove its well-posedness and propagation of chaos under mild assumptions as compared to the existing results. Speci cally, niteness of the L evy measure is not required and the coe cients are allowed to grow super-linearly in the state variable. Furthermore, we propose a tamed Euler scheme for the interacting particle system associated with the MV-SDE driven by L evy noise having super-linear growth in the state variable of the coe cients and prove that the rate of its strong convergence is arbitrarily close to 1=2. Further, we provide a tamed Milstein-type scheme for the L evy driven MV-SDE and the associated system of interacting particles where all the coe cients are growing super-linearly in the state variable. The strong rate of convergence of the proposed scheme is shown to be arbitrarily close to 1:0. For the derivation of the Milstein scheme and to show its strong rate of convergence, we provide an It^o's formula for the interacting particle system connected with the L evy driven MV-SDE. Moreover, we use the notion of Lions derivatives to examine our results. Finally, we perform some numerical experiments to demonstrate the theoretical ndings of the above results through some types of Kuramoto model, Cucker-Smale model, Ginzburg-Landau equation and double-well dynamics equation which t in our framework.