Simulation and approximation of Lévy-driven stochastic differential equations

Abstract

We consider the approximate Euler scheme for Levy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Levy measure of the driving process behaves like |z |−1−α dz near 0 , for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα . For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Levy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of order n α /(2−α ) , which is huge when α is close to 2. 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. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady [J. Komlos, P. Major and G. Tusnady, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].