Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations |
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| Institution: | 1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China;2. State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, China;1. Department of Mathematical Stochastics, University of Freiburg, Germany;2. ETH Zurich, Switzerland;1. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel;2. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368, USA;1. Dipartimento di Fisica E. Pancini, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy;2. INFN Sezione di Napoli, I-80126 Napoli, Italy |
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| Abstract: | In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented. |
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| Keywords: | Parametrix Stochastic differential equations Expansions Monte Carlo methods |
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