Abstract

A class of finite difference and finite element approximations are considered for (possibly) degenerate parabolic stochastic PDEs. Sufficient conditions are presented which ensure that the approximations admit power series expansions in terms of parameters corresponding to the mesh of the schemes. Hence, an implementation of Richardson’s extrapolation shows that the accuracy in supremum norms of suitable mixtures of approximations, corresponding to different parameters, can be as high as we wish, provided appropriate regularity conditions are satisfied. The results are applied in nonlinear filtering problems of partially observed diffusion processes. The talk is based on recent joint results with Nicolai Krylov on accelerated finite difference schemes, and joint results with Annie Millet on accelerated finite element approximations.