The Euler-Maruyama method for invariant measures of McKean-Vlasov stochastic differential equations

TL;DR

This paper studies how the Euler-Maruyama method can approximate invariant measures of McKean-Vlasov SDEs, proving convergence and uniqueness under certain conditions.

Contribution

It establishes the convergence of the EM scheme to the invariant measure and shows the particle system shares these properties, advancing numerical analysis of McKean-Vlasov SDEs.

Findings

The EM scheme converges to the invariant measure under a monotonicity condition.

The numerical solution admits a unique invariant measure with a quantifiable convergence rate.

The particle system also has a unique invariant measure with similar convergence properties.

Abstract

This paper investigates the approximation of invariant measures for McKean-Vlasov stochastic differential equations (SDEs) using the Euler-Maruyama (EM) scheme under a monotonicity condition. Firstly, the convergence of the numerical solution from the EM scheme to its continuous-time counterpart is established. Secondly, we show that the numerical solution admits a unique invariant measure and derive its convergence rate under the Wasserstein metric. In parallel, it is demonstrated that the associated particle system also possesses these properties.