Finite Difference Approximation for Linear Stochastic Partial Differential Equations with Method of Lines

A stochastic partial differential equation, or SPDE, describes the dynamics of a stochastic process defined on a space-time continuum. This paper provides a new method for solving SPDEs based on the method of lines (MOL). MOL is a technique that has largely been used for numerically solving deterministic partial differential equations (PDEs). MOL works by transforming the PDE into a system of ordinary differential equations (ODEs) by discretizing the spatial dimension of the PDE. The resulting system of ODEs is then solved by application of either a finite difference or a finite element method. This paper provides a proof that the MOL can be used to provide a finite difference approximation of the boundary value solutions for two broad classes of linear SPDEs, the linear elliptic and parabolic SPDEs.