Lévy processes, the Wiener-Hopf factorisation and applications: part III

Lévy processes, the Wiener-Hopf factorisation and applications: part II

Probability Theory Exponential Functional of Lévy Processes: Generalized Weierstrass Products and Wiener-hopf Factorization

In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalized Weierstrass products. As by-product, we obtain a multiplicative Wiener-Hopf factorization generalizing previous results obtained by the authors in [14] as well as smoothness properties of its distribution. Résumé. Fonctionnelle exponentielle des processus de Lé...

On maxima and ladder processes for a dense class of Lévy processes

Consider the problem to explicitly calculate the law of the first passage time T (a) of a general Lévy process Z above a positive level a. In this paper it is shown that the law of T (a) can be approximated arbitrarily closely by the laws of T (a), the corresponding first passages time for X, where (X)n is a sequence of Lévy processes whose positive jumps follow a phase-type distribution. Subse...

Lévy processes, the Wiener-Hopf factorisation and applications: part I

Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process. A real-valued process B = {B t : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold: (i) The paths of B are P-almost surely continuous. (ii) P(B 0 = 0) = 1. (iii) For 0 ≤ s ≤ t, B t − B s is equal in distribution to B t−s. (iv) For 0 ≤ ...

Wiener-Hopf factorization for a family of Lévy processes related to theta functions

In this paper we study the Wiener–Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévymeasure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener–Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation fo...