On reflected L\'evy processes with collapse

TL;DR

This paper analyzes a reflected Lévy process with random collapses at Poisson times, deriving its properties in general and specific cases like Brownian motion and compound Poisson processes with exponential jumps.

Contribution

It introduces a model combining reflected Lévy processes with random fractional collapses and provides detailed analysis for spectrally positive, Brownian, and compound Poisson cases.

Findings

Derived explicit formulas for the process in special cases

Characterized the distribution of the process after collapses

Extended classical reflected Lévy process theory to include collapses

Abstract

We consider a L\'evy process reflected at the origin with additional i.i.d. collapses that occur at Poisson epochs, where a collapse is a jump downward to a state which is a random fraction of the state just before the jump. We first study the general case, then specialize to the case where the L\'evy process is spectrally positive and finally we specialize further to the two cases where the L\'evy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.