On entire moments of self-similar Markov processes

TL;DR

This paper revisits and extends explicit formulas for the entire moments of positive self-similar Markov processes that jump downward and avoid hitting zero, using a recent stochastic differential equation approach.

Contribution

It provides a new proof and extension of the moment formulas for certain self-similar Markov processes using jump-type SDEs.

Findings

Reproved existing moment formulas for pssMps.

Extended formulas to broader classes of processes.

Validated approach with stochastic differential equations.

Abstract

It has been shown by Bertoin and Yor (2002) that the law of positive self-similar Markov processes (pssMps) that only jump downwards and do not hit zero in finite time are uniquely determined by their entire moments for which explicit formulas have been derived. We use a recent jump-type stochastic differential equation approach to reprove and to extend their formulas.