An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises (Journal of Computational Mathematics, 2024)
In this paper, we develop and analyze a finite difference method for linear second-orderstochastic boundary-value problems (SBVPs) driven by additive white noises. First weregularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs. We prove that the solution of the SBVP with regularized noiseconverges to the solution of the original SBVP with convergence order $\mathcal{O}(h)$ in the mean-square sense. To obtain a numerical solution, we apply the finite difference method to thestochastic BVP whose noise is piecewise constant approximation of the original noise. Theapproximate SBVP with regularized noise is shown to have better regularity than the original problem, which facilitates the convergence proof for the proposed scheme. Convergenceanalysis is presented based on the standard finite difference method for deterministic problems. More specifically, we prove that the finite difference solution converges at $\mathcal{O}(h)$ in themean-square sense, when the second-order accurate three-point formulas to approximatethe first and second derivatives are used. Finally, we present several numerical examplesto validate the efficiency and accuracy of the proposed scheme.
Keywords: Boundary-value problems; Finite-difference method; Additive white noise; Wiener process; Mean-square convergence; Wong-Zakai approximation.