ANTITHETIC MULTILEVEL SAMPLING METHOD FOR NONLINEAR FUNCTIONALS OF MEASURE

• 成果类型: Article
• 署名作者: Szpruch, Lukasz; Tse, Alvin
• 署名单位: University of Edinburgh
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1614
• 发表日期: 2021
• 页码: 1100-1139
• 关键词: monte-carlo methods; mckean-vlasov; propagation; approximation; CONVERGENCE; chaos; limit

摘要

Let mu is an element of P-2(R-d), where P-2(R-d) denotes the space of square integrable probability measures, and consider a Borel-measurable function Phi : P-2(R-d) -> R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for Phi(mu), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws mu(N) = 1/N Sigma(N)(i =1) delta X-i, where (a) (X-i)(i =1)(N) is a sequence of i.i.d. samples from mu or (b) (X-i)(i =1)(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 L-2 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.