Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes

Sylvain Rubenthaler* and Magnus Wiktorsson**

* Laboratoire de probabilités et modèles aléatoires (UMR 7599), Paris VI, 4, Place Jussieu, 75252, Paris, Cédex 05, France
** Centre for Mathematical Sciences, Mathematical Statistics, Lund University, Box 118, SE-221 00, Lund, Sweden

Received 22 January 2003;  revised 19 June 2003;  accepted 1 July 2003. ; Available online 22 July 2003.


Abstract

We consider the Euler approximation of stochastic differential
equations (SDEs) driven by Lévy processes in the case where we cannot
simulate the increments of the driving process exactly. In some cases,
where the driving process Y is a subordinated stable process, i.e.,
Y=Z(V) with V a subordinator and Z a stable process, we propose an
approximation Y by Z(Vn) where Vn is an approximation of V. We then
compute the rate of convergence for the approximation of the solution
X of an SDE driven by Y using results about the stability of SDEs.

Author Keywords: Stochastic differential equation; Numerical
approximation; Convergence rate; Lévy process; Shot noise
representation; Subordination

Mathematical subject codes: primary 60H10; 65C30; secondary 60G51; 60F17