Convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation

Abstract

This paper studies the convergence of a spatial semidiscretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semidiscretization uses the standard continuous piecewise linear finite element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semidiscretization. Thirdly, an application of the theoretical results to a stochastic linear quadratic control problem is presented.

Temporal semi-discretizations of a backward semilinear stochastic evolution equation

“…Remark 5.5. Our previous work [29] has analyzed a spatial semi-discretization for a stochastic linear quadratic control problem. For a special case of problem (5.7), Li and Zhou [30] obtained the temporal accuracy O(τ 1/2 ) for rough data.…”

“…Wang [42] analyzed a discretization for a backward semilinear stochastic parabolic equation; since this discretization uses the eigenvectors of the Laplace operator, its application appears to be limited. Recently, Li and Xie [29] analyzed a spatial semi-discretization for a backward semilinear stochastic parabolic equation with general filtration, using the standard piecewise linear finite element method. To our best knowledge, no numerical analysis of temporal semi-discretizations is available for a backward semilinear stochastic evolution equation in an infinite-dimensional Hilbert space.…”