Semi-discretization of stochastic partial differential equations on R1 by a finite-difference method

The paper concerns finite-difference scheme for the approximation of partial differential equations in R 1 , with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in R 1 with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted Lp-theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.

[1]  Nicolai V. Krylov,et al.  On Lp-theory of stochastic partial di6erential equations in the whole space , 1996 .

[2]  Jessica G. Gaines,et al.  Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations , 2001, Math. Comput..

[3]  I. Gyöngy.  Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White Noise I , 1998 .

[4]  J. Bennaton,et al.  Discrete time Galerkin approximations to the nonlinear filtering solution , 1985 .

[5]  Boris Rozovskii,et al.  Stochastic partial differential equations and diffusion processes , 1982 .

[6]  René Carmona,et al.  Stochastic Partial Differential Equations: Six Perspectives , 1998 .

[7]  N. Bellomo,et al.  Stochastic partial differential equations in continuum physics—on the foundations of the stochastic interpolation method for ITO's type equations , 1989 .

[8]  M. Piccioni,et al.  Semi-discretization of stochastic partial differential equations on rdby a Finite-element Technique A. Germani , 1988 .

[9]  N. Krylov.  Introduction to the theory of diffusion processes , 1994 .

[10]  E. Helfand.  Numerical integration of stochastic differential equations , 1979, The Bell System Technical Journal.

[11]  P. Kloeden,et al.  Numerical Solutions of Stochastic Differential Equations , 1995 .

[12]  R. Glowinski,et al.  Approximation of some stochastic differential equations by the splitting up method , 1992 .

[13]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[14]  B. Rozovskii.  Stochastic Evolution Systems , 1990 .

[15]  A. Germani.  Semi-discretization of stochastic partial diffential equations on Rd by a finite-element technique , 1988 .

[16]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[17]  Nicola Bellomo,et al.  Nonlinear Stochastic Evolution Problems in Applied Sciences , 1992 .

[18]  N. Krylov,et al.  Lectures on Elliptic and Parabolic Equations in Holder Spaces , 1996 .

[19]  J. Gaines.  Stochastic Partial Differential Equations: Numerical experiments with S(P)DE's , 1995 .