Two kinds of conditionings for stable L\'evy processes

TL;DR

This paper explores two types of conditionings for one-dimensional stable Lévy processes using $h$-transforms of excursion measures, focusing on processes that stay positive or avoid the origin.

Contribution

It introduces a novel approach to condition stable Lévy processes via $h$-transforms, expanding understanding of their behavior under specific constraints.

Findings

Characterization of stable Lévy processes conditioned to stay positive.

Analysis of processes conditioned to avoid the origin.

Development of $h$-transform techniques for these conditionings.

Abstract

Two kinds of conditionings for one-dimensional stable L\'evy processes are discussed via $ h $-transforms of excursion measures: One is to stay positive, and the other is to avoid the origin.