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June 2021 Antithetic multilevel sampling method for nonlinear functionals of measure
Łukasz Szpruch, Alvin Tse
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Ann. Appl. Probab. 31(3): 1100-1139 (June 2021). DOI: 10.1214/20-AAP1614

Abstract

Let μP2(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=1Ni=1NδXi, 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.

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

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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

Information

Received: 1 May 2019; Revised: 1 July 2020; Published: June 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.1214/20-AAP1614

Subjects:
Primary: 60H10 , 60K35 , 65C35

Keywords: antithetic multi-level Monte Carlo estimator , McKean–Vlasov SDEs , propagation of chaos , Wasserstein calculus

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 3 • June 2021
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