Weak Convergence of a Sequence of Semimartingales to a Diffusion with Discontinuous Drift and Diffusion Coefficients

Abstract

This paper presents a set of sufficient conditions for a sequence of semimartingales to converge weakly to a solution of a stochastic differential equation (SDE) with discontinuous drift and diffusion coefficients. This result is closely related to a well-known weak-convergence theorem due to Liptser and Shiryayev (see [27]) which proves the weak convergence to a solution of a SDE with continuous drift and diffusion coefficients in the Skorokhod–Lindvall J1-topology. The goal of this paper is to obtain a stronger result in order to solve outstanding problems in the area of large-scale queueing networks – in which the weak convergence of normalized queueing length is a solution of a SDE with discontinuous coefficients. To do this we need to make the stronger assumptions: (1) replacing the convergence in probability of the triplets of a sequence of semimartingales in the original Liptser and Shiryayev's theorem by stronger convergence in L2, (2) assuming the diffusion coefficient is coercive, and (3) assuming the discontinuity sets of the coefficients of the limit diffusion processs are of Lebesgue measure zero.