Some explicit identities associated with positive self-similar Markov processes

TL;DR

This paper derives explicit formulas for the laws of certain positive self-similar Markov processes, focusing on their exit times, exponential functionals, and hitting probabilities, based on special classes of Lévy processes with specific measures.

Contribution

It provides explicit computations of exit time laws, exponential functionals, and hitting probabilities for a class of Lévy processes related to self-similar Markov processes, extending prior theoretical results.

Findings

Explicit law of Lévy processes at first exit time

Distribution of exponential functionals

First hitting time probabilities

Abstract

We consider some special classes of L\'evy processes with no gaussian component whose L\'evy measure is of the type $\pi(dx)=e^{\gamma x}\nu(e^x-1) dx$, where $\nu$ is the density of the stable L\'evy measure and $\gamma$ is a positive parameter which depends on its characteristics. These processes were introduced in \cite{CC} as the underlying L\'evy processes in the Lamperti representation of conditioned stable L\'evy processes. In this paper, we compute explicitly the law of these L\'evy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.