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. Keywords: stochastic differential equations, discontinuous drift, degenerate diffusion, existence and uniqueness of solutions, numerical methods for stochastic differential equations, strong convergence rate Mathematics Subject Classification (2010): 60H10, 65C30, 65C20 (Primary), 65L20 (Secondary)

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We prove strong convergence of order for arbitrarily small of the Euler–Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler–Maruyama scheme and another numerical method, which is constructed by applying the Euler–Maruyama scheme to a transformation of the SDE we aim to solve.

In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler-Maruyama scheme for such a class of SDEs. We prove convergence of order 1/2. Finally, we present numerical examples.

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of highdimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group.We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.