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 existence and uniqueness as well as 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 and we present an application from stochastic optimal control.
"A strong order $1/2$ method for multidimensional SDEs with discontinuous drift." Ann. Appl. Probab. 27 (4) 2383 - 2418, August 2017. https://doi.org/10.1214/16-AAP1262