Simulation and approximation of Lévy-driven stochastic differential equations

LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France; nicolas.fournier@univ-paris12.fr

Received: 10 April 2009
Revised: 11 September 2009

Abstract

We consider the approximate Euler scheme for Lévy-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 Lévy measure of the driving process behaves like |z|^−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order n^α. 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 Lévy 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 Lévy-driven S.D.E. by a Brownian S.D.E. when the Lévy 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 Poincaré Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlós-Major-Tsunády [J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].