Convergence of a spatial semidiscretization for a backward semilinear stochastic parabolic equation

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 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 result to a stochastic linear quadratic control problem is presented.