Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

Annika Lang and Andreas Petersson

Mathematics and Computers in Simulation (MATCOM), 2018, vol. 143, issue C, 99-113

Abstract: The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Yn is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Yn to Y in terms of the error |E[Y−Yn]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y]−EN[Yn]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |EN[Y−Yn]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y−Yn]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations.

Keywords: (multilevel) Monte Carlo methods; Variance reduction techniques; Stochastic partial differential equations; Weak convergence; Upper and lower error bounds (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:143:y:2018:i:c:p:99-113

DOI: 10.1016/j.matcom.2017.05.002

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