Simulation of stochastic integrals with respect to Lévy processes of type G

Magnus Wiktorsson

Centre for Mathematical Sciences, Lund University, Box 118, 221 00, Lund, Sweden

Received 22 August 2001;  revised 7 March 2002;  accepted 19 March 2002.  Available online 17 April 2002.


Abstract

We study the simulation of stochastic processes defined as stochastic
integrals with respect to type G Lévy processes for the case where it
is not possible to simulate the type G process exactly. The type G
Lévy process as well as the stochastic integral can on compact
intervals be represented as an infinite series. In a practical
simulation we must truncate this representation. We examine the
approximation of the remaining terms with a simpler process to get an
approximation of the stochastic integral. We also show that a
stochastic time change representation can be used to obtain an
approximation of stochastic integrals with respect to type G Lévy
processes provided that the integrator and the integrand are
independent.

Author Keywords: Type G distribution; Stochastic integral; Variance
mixture; Lévy process; Shot noise representation; Stochastic time
change; Subordination

PACS classification codes: 60G51; 60H05; 60E07