A finite difference method for stochastic nonlinear second-order boundary-value problems driven by additive noises

In this paper, we present a finite difference method for stochastic nonlinear secondorder boundary-value problems (BVPs) driven by additive noises. We first approximate the white noise process with its piecewise constant approximation to obtain an approximate stochastic BVP. The solution to the new BVP is shown to converge to the solution of the original BVP at O(h) in the mean-square sense. The approximate BVP is shown to have certain regularity properties which are not true in general for the solution to the original stochastic BVP. The standard finite difference method for deterministic BVPs is then applied to approximate the solution of the new stochastic BVP. Convergence analysis is presented for the numerical solution based on the standard finite difference method. We prove that the finite difference solution converges to the solution to the original stochastic BVP at O(h) in the mean-square sense. Finally, we perform several numerical examples to validate the theoretical results.