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

TL;DR

This paper extends Rio's multidimensional central limit bound to higher dimensions with dimension-independent moments, and develops two methods for simulating multidimensional Le9vy SDEs, enhancing understanding and computational approaches.

Contribution

It generalizes Rio's central limit bound to multiple dimensions with a novel approach that requires dimension-independent moments, and introduces two new simulation methods for multidimensional Le9vy SDEs.

Findings

Extended Rio's bound to multidimensional case with dimension-independent moments

Developed two approaches for simulating multidimensional Le9vy SDEs

Provided theoretical foundation for better approximation and simulation of Le9vy processes

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.