Abstract
Let , where denotes the space of square integrable probability measures, and consider a Borel-measurable function . In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for , which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws , where (a) is a sequence of i.i.d. samples from μ or (b) 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 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.
Funding Statement
This was work has been supported by The Alan Turing Institute under the Engineering and Physical Sciences Research Council Grant EP/N510129/1.
Citation
Łukasz Szpruch. Alvin Tse. "Antithetic multilevel sampling method for nonlinear functionals of measure." Ann. Appl. Probab. 31 (3) 1100 - 1139, June 2021. https://doi.org/10.1214/20-AAP1614
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