Stability and error analysis of an implicit Milstein finite difference scheme for a two-dimensional Zakai SPDE

TL;DR

This paper introduces an implicit finite difference scheme for a 2D Zakai SPDE, demonstrating stability and convergence through Fourier analysis, with numerical tests confirming theoretical results.

Contribution

It presents a novel implicit Milstein finite difference scheme for 2D Zakai SPDEs, proving stability and convergence in mean-square sense.

Findings

Scheme is mean-square stable under certain conditions.

Convergence is first order in time and second order in space.

Numerical tests confirm theoretical stability and convergence.

Abstract

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.