Higher order spatial approximations for degenerate parabolic stochastic partial differential equations
Provides a theoretical convergence acceleration result for numerical solutions of degenerate parabolic SPDEs, relevant to nonlinear filtering.
The paper develops a finite difference scheme for degenerate parabolic SPDEs and shows that spatial convergence can be accelerated to arbitrarily high order via extrapolation.
We consider an implicit finite difference scheme on uniform grids in time and space for the Cauchy problem for a second order parabolic stochastic partial differential equation where the parabolicity condition is allowed to degenerate. Such equations arise in the nonlinear filtering theory of partially observable diffusion processes. We show that the convergence of the spatial approximation can be accelerated to an arbitrarily high order, under suitable regularity assumptions, by applying an extrapolation technique.