Fluctuation theory and exit systems for positive self-similar Markov processes

TL;DR

This paper develops a detailed fluctuation theory for positive self-similar Markov processes, constructing local times, exit systems, and ladder processes, and proves convergence results as the process starts near zero.

Contribution

It introduces a new framework for analyzing excursions and ladder processes of positive self-similar Markov processes, addressing open problems in the field.

Findings

Constructed a local time for the set of times when the process reaches its past supremum.

Described an exit system for excursions out of the past supremum.

Proved finite-dimensional convergence of the ladder process as the starting point approaches zero.

Abstract

For a positive self-similar Markov process, X, we construct a local time for the random set, $\Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $\Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L\'{e}vy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L\'{e}vy process oscillates.