A Multi-Dimensional Central Limit Bound and its Application to the Euler Approximation of L\'evy SDEs

Abstract

Rio gave a concise bound for the central limit theorem in the Vaserstein distances, which is a ratio between some higher moments and some powers of the variance. As a corollary, it gives an estimate for the normal approximation of the small jumps of the L\'evy processes, and Fournier applied that to the Euler approximation of L\'evy-driven stochastic differential equations. However both results are restricted to the one-dimensional case. It will be shown in this article that, following Davie's idea, one can generalise Rio's result to multidimensional case, and the number of moments required is independent of the dimension. Also two different approaches are developed to simulate multidimensional L\'evy SDEs: one as a corollary to the central limit bound, the other directly derived from the L\'evy-Khinchine formula.