Simulation and approximation of Levy-driven stochastic differential equations

TL;DR

This paper develops a numerical scheme for simulating Levy-driven stochastic differential equations by exactly simulating large jumps and approximating small jumps with Gaussian variables, providing error estimates and studying Brownian approximations.

Contribution

It introduces a new simulation method for Levy-driven SDEs with error bounds, especially effective near singular Levy measures, and analyzes Brownian approximations for Levy processes without large jumps.

Findings

The scheme accurately simulates Levy-driven SDEs with controlled error.

Error estimates remain reasonable even with singular Levy measures.

Approximation of Levy-driven SDEs by Brownian SDEs is effective when no large jumps occur.

Abstract

We consider the problem of the simulation of Levy-driven stochastic differential equations. It is generally impossible to simulate the increments of a Levy-process. Thus in addition to an Euler scheme, we have to simulate approximately these increments. We use a method in which the large jumps are simulated exactly, while the small jumps are approximated by Gaussian variables. Using some recent results of Rio about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady, we derive an estimate for the strong error of this numerical scheme. This error remains reasonnable when the Levy measure is very singular near 0, which is not the case when neglecting the small jumps. In the same spirit, we study the problem of the approximation of a Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large jumps.