Monte Carlo approximation for stochastic differential equations with interaction

Abstract

We study a Monte Carlo approximation scheme for stochastic differential equations with interaction. The initial measure is replaced by its empirical approximation constructed from independent samples, and the corresponding equation is discretized in time by the Euler-Maruyama scheme. For initial measures with finite moments of order q>2, we derive a quantitative estimate for the empirical approximation in the Wasserstein distance of order two. Combining this estimate with stability estimates for stochastic flows with interaction, we obtain a convergence bound for the total Euler-Maruyama-Monte Carlo approximation. We also compare the particle-based approximation with the deterministic tensor-grid construction used in earlier work.