Higher order spatial approximations for degenerate parabolic stochastic partial differential equations

TL;DR

This paper develops a high-order spatial approximation scheme for degenerate second-order parabolic stochastic PDEs, enabling faster convergence through extrapolation under certain regularity conditions.

Contribution

It introduces an extrapolation technique to accelerate convergence of finite difference schemes for degenerate stochastic PDEs, a novel approach in this context.

Findings

Convergence can be accelerated to arbitrarily high order.

The scheme is effective under suitable regularity assumptions.

Applicable to equations in nonlinear filtering theory.

Abstract

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.