Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions

2007/12/07 by Pierre Patie, Patie, P. · 1 citation
Economics, Econometrics and Finance · Mathematics · #60B52. #60G18 #60G51 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0712.1115

openalex publication_date 2007/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We first introduce and derive some basic properties of a two-parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian motion with drifts. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way we also characterize explicitly the density of the semi-groups of the family of self-similar continuous state branching processes with immigration.

Citations

• The Asymptotic Expansion of Generalized Hypergeometric Functions

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• On the Lamperti stable processes

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