Antithetic multilevel sampling method for nonlinear functionals of measure

Abstract

Let $\mathit{\mu }\in {\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}})$, where ${\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}})$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $\mathrm{\Phi }:{\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}})\to \mathbb{R}$. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for $\mathrm{\Phi }(\mathit{\mu })$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws ${\mathit{\mu }}^{\mathit{N}}=\frac{1}{\mathit{N}}{\sum }_{\mathit{i}=1}^{\mathit{N}}{\mathit{\delta }}_{{\mathit{X}}_{\mathit{i}}}$, where (a) ${({\mathit{X}}_{\mathit{i}})}_{\mathit{i}=1}^{\mathit{N}}$ is a sequence of i.i.d. samples from μ or (b) ${({\mathit{X}}_{\mathit{i}})}_{\mathit{i}=1}^{\mathit{N}}$ 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 strong propagation of chaos, as well as an ${\mathit{L}}^2$ estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.