WEIGHTED MULTILEVEL LANGEVIN SIMULATION OF INVARIANT MEASURES

• 成果类型: Article
• 署名作者: Pages, Gilles; Panloup, Fabien
• 署名单位: Sorbonne Universite; Universite d'Angers
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1364
• 发表日期: 2018
• 页码: 3358-3417
• 关键词: richardson-romberg extrapolation; recursive computation; approximation; diffusion; equation; limit

摘要

We investigate a weighted multilevel Richardson-Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in [Bernoulli 23 (2017) 2643-2692] for regular Monte Carlo simulation. In a first result, we prove under weak confluence assumptions on the diffusion, that for any integer R >= 2, the procedure allows us to attain a rate n(R/2R+1) whereas the original algorithm convergence is at a weak rate n(1/3). Furthermore, this is achieved without any explosion of the asymptotic variance. In a second part, under stronger confluence assumptions and with the help of some second-order expansions of the asymptotic error, we go deeper in the study by optimizing the choice of the parameters involved by the method. In particular, for a given epsilon > 0, we exhibit some semi-explicit parameters for which the number of iterations of the Euler scheme required to attain a mean-squared error lower than epsilon(2) is about epsilon(-2) log(epsilon(-1)). Finally, we numerically test this multilevel Langevin estimator on several examples including the simple one-dimensional Ornstein-Uhlenbeck process but also a high dimensional diffusion motivated by a statistical problem. These examples confirm the theoretical efficiency of the method.