CENTRAL LIMIT THEOREM FOR THE ANTITHETIC MULTILEVEL MONTE CARLO METHOD

• 成果类型: Article
• 署名作者: Ben Alaya, Mohamed; Kebaier, Ahmed; Ngo, Thi Bao Tram
• 署名单位: Universite de Rouen Normandie; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1726
• 发表日期: 2022
• 页码: 1970-2027
• 关键词: sdes; scheme

摘要

In this paper, we give a natural extension of the antithetic multilevel Monte Carlo (MLMC) estimator for a multidimensional diffusion introduced by Giles and Szpruch (Ann. Appl. Probab. 24 (2014) 1585-1620) by considering the permutation between m Brownian increments, m >= 2, instead of using two increments as in the original paper. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Levy area and its s-antithetic version build on the finer grid, and on the other hand, the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Levy area and its s-antithetic version stably converges in distribution with a rate of order 1/2. More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see, e.g., Jacod (In Seminaire de Probabilites, XXXI (1997) 232-246 Springer). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the antithetic MLMC estimator. The time complexity of the algorithm is analyzed.