A Ciesielski-Taylor type identity for positive self-similar Markov processes

TL;DR

This paper provides a simplified proof of a generalized Ciesielski-Taylor identity for positive self-similar Markov processes of spectrally negative type, unifying previous results through a novel transformation and fluctuation theory.

Contribution

It introduces a new transformation of Laplace exponents and offers a straightforward proof that encompasses all known identities in this class.

Findings

Unified Ciesielski-Taylor identity for spectrally negative processes

New transformation of Laplace exponents preserves process class

Simplified proof using classical and recent fluctuation identities

Abstract

The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly a new transformation which maps a subset of the family of Laplace exponents of spectrally negative L\'evy processes into itself. Secondly some classical features of fluctuation theory for spectrally negative L\'evy processes as well as more recent fluctuation identities for positive self-similar Markov processes.