A discontinuous Galerkin method for systems of stochastic differential equations with applications to population biology, finance, and physics

In this paper, we propose a discontinuous Galerkin (DG) method for systems of stochastic differential equations (SDEs) driven by m-dimensional Brownian motion. We first construct a new approximate system of SDEs on each element using whose converges to the solution of the original system. The new system is then discretized using the standard DG method for deterministic ordinary differential equations (ODEs). For the case of additive noise, we prove that the proposed scheme is convergent in the mean-square sense. Our numerical experiments suggest that our results hold true for the case of multiplicative noise as well. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, the proposed scheme is illustrated by considering different examples arising in population biology, physics, and mathematical finance.