ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE

• 来源期刊: 计算数学（英文版）
• 关键词: Parabolic stochastic partial differential equations Multilevel Monte Carlo Multi-index Monte Carlo Stochastic finite differences Zakai equation
• 年，卷（期）: 2018,（2）
• 页码范围: 202-236
• 页数: 35页
• 语种: 英文
• DOI: 10.4208/jcm.1612-m2016-0681

摘要

In this article,we propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type.We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012),and find,by means of Fourier analysis,that the MIMC method:(i) has suboptimal complexity of O(ε-2| logε|3) for a root mean square error (RMSE) ε if the same spatial discretisation as in the MLMC method is used;(ii) has a better complexity of O(ε-2| logε|) if a carefully adapted discretisation is used;(iii) has to be adapted for non-smooth functionals.Numerical tests confirm these findings empirically.