Sample path behaviors of L\'{e}vy processes conditioned to avoid zero

TL;DR

This paper investigates the detailed sample path behaviors of Lévy processes conditioned to avoid zero, focusing on their long-term and short-term dynamics, and builds on Takeda-Yano's work on their limiting processes via Doob's h-transform.

Contribution

It provides a detailed analysis of the sample path behaviors of the limit processes conditioned to avoid zero, extending Takeda-Yano's results.

Findings

Characterization of long-term sample path behaviors

Analysis of short-term dynamics of conditioned Lévy processes

Insights into the influence of different random clocks on process limits

Abstract

Takeda-Yano determined the limit of L\'{e}vy processes conditioned to avoid zero via various random clocks in terms of Doob's $h$-transform, where the limit processes may differ according to the choice of random clocks. The purpose of this paper is to investigate sample path behaviors of the limit processes in long time and in short time.