The convergence of the EM scheme in empirical approximation of invariant probability measure for McKean-Vlasov SDEs

Based on the assumption of the existence and uniqueness of the invariant measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a self-interacting process that depends only on the current and historical information of the solution is constructed for MV-SDEs. The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric. Furthermore, under the condition of linear growth, an EM scheme whose uniformly 1/2-order convergence rate with respect to time is obtained is constructed for the self-interacting process. Then, the convergence rate between the weighted empirical measure of the EM numerical solution of the self-interacting process and the invariant measure of MV-SDEs is derived. Moreover, the convergence rate between the averaged weighted empirical measure of the EM numerical solution of the corresponding multi-particle system and the invariant measure of MV-SDEs in the W2-Wasserstein metric is also given. In addition, the computational cost of the two approximation methods is compared, which shows that the averaged weighted empirical approximation of the particle system has a lower cost. Finally, the theoretical results are validated through numerical experiments.