L\'evy processes resurrected in the positive half-line

TL;DR

This paper studies a modified Le9vy process constrained to stay positive by removing negative jumps, analyzing its behavior at zero and providing conditions for absorption or non-absorption based on the original process.

Contribution

It characterizes the law of the resurrected Le9vy process and establishes criteria for absorption at zero in various classes of Le9vy processes.

Findings

Derived the law of the resurrected process from the original Le9vy process.

Provided conditions for absorption at zero.

Identified classes where the process is non-absorbing.

Abstract

A L\'evy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. We first describe the law of the resurrected process in terms of that of the initial L\'evy process. Then in many important classes of L\'evy processes, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial L\'evy process.