A Strong Order 1/2 Method for Multidimensional SDEs with Discontinuous Drift
For researchers working on numerical solutions of SDEs with irregular coefficients, this provides a first strong convergence result for a broad class of problems.
The paper proves existence and uniqueness for multidimensional SDEs with discontinuous drift and possibly degenerate diffusion, and presents a numerical method achieving strong order 1/2 convergence, the first such result for this general class.
In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.