On simulation from infinitely divisible distributions

Abstract

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

Invariance properties of the negative binomial Levy process and stochastic self-similarity

“…The above results lead to an approximate method of simulating gamma processes using the NBP with a small value of p, which provides an alternative over the existing methods (see, e.g., [2,17,20]). The simplest approach is to use Theorem 4.1 and approximate the gamma process by a rescaled NBP.…”

Quasi-Monte Carlo Method for Infinitely Divisible Random Vectors via Series Representations

“…To obtain a closed form, some alternative methods have been proposed, for example, the thinning method and the rejection method of [21]. Each of those methods can be considered as a special case of the so-called generalized shot noise method of [3,21], which we describe as follows. Assume that Lévy measure ν can be decomposed as…”

“…The theory of stable processes and their applications are expanded, due to LePage [16], on series representation of stable random vectors. The simulation of nonnegative infinitely divisible random variables is considered, and their series representations as a special form of generalized shot noise is developed in Bondesson [3]. The same approach is used in Rosiński [21] as a general pattern for series representations of Banach space valued infinitely divisible random vectors.…”

Modelling by Levy Processes

“…The representations are useful, in particular, for simulation of such processes and integrals, whereas direct simulation is not really practical due to the jump character of the processes. Bondesson (1982) was the first to discuss the type of approach we shall consider, more recent work being due to Marcus (1987), Rosinski (1991), Asmussen (1998, Sect. VIII.2), Wolpert and Ickstadt (1998) and Wolpert and Ickstadt (1999).…”