Convergence analysis of an explicit method and its random batch approximation for the McKean–Vlasov equations with non-globally Lipschitz conditions

¹ Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China
² Department of Mathematics, Jiangsu Second Normal University, Nanjing 210013, P.R. China
³ School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, DCI joint team, Shanghai Jiao Tong University, 200240 Shanghai, P.R. China

^* Corresponding author: 1000497941@smail.shnu.edu.cn

Received: 28 August 2022
Accepted: 30 December 2023

Abstract

In this paper, we present a numerical approach to solve the McKean–Vlasov equations, which are distribution-dependent stochastic differential equations, under some non-globally Lipschitz conditions for both the drift and diffusion coefficients. We establish a propagation of chaos result, based on which the McKean–Vlasov equation is approximated by an interacting particle system. A truncated Euler scheme is then proposed for the interacting particle system allowing for a Khasminskii- type condition on the coefficients. To reduce the computational cost, the random batch approximation proposed in [S. Jin, L. Li and J. Liu, J. Comput. Phys. 400 (2020) 108877.] is extended to the interacting particle system where the interaction could take place in the diffusion term. An almost half order of convergence is proved in L^p sense. Numerical tests are performed to verify the theoretical results.