Antithetic multilevel particle system sampling method for McKean-Vlasov SDEs

Abstract

Let $\mu \in {\mathcal{P}}_2({\mathbb{R}}^d)$, where ${\mathcal{P}}_2({\mathbb{R}}^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $\Phi :{\mathcal{P}}_2({\mathbb{R}}^d)\to \mathbb{R}$. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $\Phi (\mu )$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws ${\mu }^N=\frac{1}{N}{\sum }_{i=1}^N{\delta }_{X_i}$, where a) $(X_i{)}_{i=1}^N$ is a sequence of i.i.d samples from $\mu$ or b) $(X_i{)}_{i=1}^N$ is a system of interacting particles (diffusions) corresponding to a McKean-Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform \textit{strong propagation of chaos}, as well as an $L^2$ estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.