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 J 1-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 L 2, (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.
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Chao, YJ. Weak Convergence of a Sequence of Semimartingales to a Diffusion with Discontinuous Drift and Diffusion Coefficients. Queueing Systems 42, 153–188 (2002). https://doi.org/10.1023/A:1020105004865
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DOI: https://doi.org/10.1023/A:1020105004865