Abstract
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 L_2-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.
Highlights
The analysis and numerical computation of Zakai equations and other types of stochastic partial differential equation (SPDE) have been extensively studied in recent years
We study the mean-square stability, and the strong convergence of the second moment
We combine the scheme with an Alternating Direction Implicit (ADI) factorisation, which has been introduced in [34] for parabolic PDEs to approximately factorise the system matrix by matrices which correspond to derivatives in individual directions and which can more be inverted, while the consistency order is maintained
Summary
The analysis and numerical computation of Zakai equations and other types of stochastic partial differential equation (SPDE) have been extensively studied in recent years. Giles and Reisinger [16] use an explicit Milstein finite difference approximation to the solution of the following one-dimensional SPDE, a special case of (1.1) for d = 1 and constant coefficients, dv. Where T > 0, M is a standard Brownian motion, and μ and 0 ≤ ρ < 1 are real-valued parameters This is extended in [35] to an approximation of (1.6) with an implicit method on the basis of the σ –θ time-stepping scheme, where the finite variation parts of the double stochastic integral are taken implicit. The sharp estimates we derive give a precise description of the error given Dirac initial data This is achieved by a Fourier analysis originating from Carter and Giles in [7], where they estimated the error arising from explicit and implicit approximations of the constant-coefficient 1-d convection–diffusion equation with Dirac initial data.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
