Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

TL;DR

This paper analyzes the weak error in mean-field particle approximations of McKean-type SDEs, demonstrating an error of order 1/N + h, and explores the effectiveness of antithetic sampling through numerical experiments.

Contribution

It provides a theoretical error bound for particle approximations of McKean SDEs and evaluates antithetic sampling's efficiency in this context.

Findings

Weak error is O(1/N + h) for the approximation.

Numerical experiments confirm theoretical error bounds.

Antithetic sampling improves efficiency in mean-field simulations.

Abstract

In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.