The discontinuous Galerkin method for stochastic differential equations driven by additive noises

In this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when p-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.