Remarks on Levy Process Simulation

Algorithms for simulation of a Lévy process X(t) are discussed, with particular emphasis on two algorithms approximating jumps that are in some sense small. One is classical, defining small jumps as those of small absolute value. The other one appears to be new and relies on an completely monotone structure of the Lévy density n(x). One then truncates the representing measure of n(X) to [0,A], meaning that jumps of mean <1/A are left out. In both algorithms, the large jump part is simulated as compound Poisson and the small jumps are approximated. The standard choice of such an approximation is normal with the same mean and variance, but we also consider gamma approximations in two variants, and show that in some cases these perform substantially better. Other algorithms are briefly surveyed and we sketch a new one for simulation of a tempered stable (CGMY) process with infinite variation.