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

TL;DR

This paper introduces a numerical method for McKean-Vlasov equations with non-globally Lipschitz coefficients, establishing convergence and extending random batch approximation to reduce computational costs.

Contribution

It develops a truncated Euler scheme and extends the random batch approximation to the interacting particle system for McKean-Vlasov equations, with proven convergence.

Findings

Almost half order of convergence in $L^p$ sense.

Numerical results verify theoretical convergence.

Extension of random batch method to diffusion interactions.

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 [Jin et al., J. Comput. Phys., 400(1), 2020] 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.