Antithetic multilevel sampling method for nonlinear functionals of measure

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 takes as input empirical laws μN=1N∑ i=1NδX i, (a) (Xi)i=1N is sequence i.i.d. samples from μ or (b) system interacting particles (diffusions) corresponding to McKean–Vlasov stochastic differential equation (McKV-SDE). Each case requires separate analysis. For mean-field particle system, also law induced by its Euler discretisation gives fully implementable algorithm. As by-products our analysis, establish dimension-independent rate uniform strong propagation chaos, well L2 estimate difference random variables general functionals defined on measures.