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

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 Lp sense. Numerical tests are performed to verify the theoretical results.