In this article, we propose an implicit finite difference scheme for a two-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. The scheme is based on a Milstein approximation to the stochastic integral and an alternating direction implicit (ADI) discretisation of the elliptic term. We prove its mean-square stability and convergence in L2 of first order in time and second order in space, by Fourier analysis, in the presence of Dirac initial data. Numerical tests confirm these findings empirically.
Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative
Deep learning based numerical approximation algorithms for stochastic partial differential equations
Department of Mathematics, ETH Zurich（苏黎世联邦理工学院数学系）；School of Data Science and School of Artificial Intelligence, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenhen)（香港中文大学（深圳）数据科学学院和人工智能学院）；Applied Mathematics: Institute for Analysis and Numerics, University of Münster（乌尔姆大学应用数学研究所）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学数学科学系）
