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

This article proposes an implicit finite difference scheme for a two-dimensional parabolic stochastic partial differential equation of Zakai type. The scheme is based on a Milstein approximation to the stochastic integral and an alternating direction implicit discretisation of the elliptic term. Mean-square stability and L2 -convergence of first order in time and second order in space are proven by Fourier analysis, in the presence of Dirac initial data. Numerical tests confirm these findings empirically.