Numerical Path Integration for Lévy Driven Stochastic Differential Equations

Abstract

Some theory on Lévy processes and stochastic differential equations driven by Lévy processes is reviewed. Inverse Fast Fourier Transform routines are applied to compute the density of the increments of Lévy processes. We look at exact and approximate path integration operators to compute the probability density function of the solution process of a given stochastic differential equation. The numerical path integration method is shown to converge under the transition kernel backward convergence assumption. The numerical path integration method is applied on several examples with non-Brownian driving noises and nonlinearities, and shows satisfactory results. In the case when the noise is of additive type, a general code written for Lévy driving noises specified by the Lévy-Khintchine formula is described. A preliminary result on path integration in Fourier space is given.