Title    name Stochastic Partial Differential Equations and the Related Fields.     Title Accelerated spatial approximations for time discretized SPDE   Speaker Eric J. Hall   Date 2012-06-25   Host   Place NIMS   File     Abstract : We investigate numerical solutions of the Cauchy problem for linear second order parabolic stochastic partial differential equations (SPDE) defined on the whole space. Such SPDE arise in applications of the nonlinear filtering theory of partially observable diffusion processes where solutions are desired in real-time. Thus there is a keen interest in developing accurate numerical schemes for their solutions. We consider finite difference approximations in uniform grids in time and space and give sufficient conditions for accelerating the strong convergence with respect to the spatial approximation to higher order accuracy by an extrapolation technique. This is done by first proving the existence of an expansion in powers of the spatial discretization parameter for the solution to our space-time scheme, provided appropriate regularity conditions are satisfied. Hence we apply Richardson's method and show that suitable mixtures of the spatial difference approximations at different mesh sizes converge to the time discretized solution with arbitrarily high accuracy. This work extends the results of Gyöngy and Krylov [SIAM J. Math. Anal., 42 (2010), pp. 2275--2296] to schemes that discretize in time as well as space.

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