Antithetic multilevel sampling method for nonlinear functionals of measure

Abstract

Let μ∈ P2(Rd), where P2(Rd) denotes the space of square integrable probability measures, and consider a Borel-measurable function Φ:P2(Rd)→R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for Φ(μ), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws μN=1N∑ i=1NδX i, where (a) (Xi)i=1N is a sequence of i.i.d. samples from μ or (b) (Xi)i=1N 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 L2 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.