Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On reflected Lévy processes with collapse

Onno Boxma, Offer Kella, David Perry

TL;DR

We study a reflected Lévy process with random, proportional collapses occurring at Poisson times and derive a Lindley-type autoregressive structure for the pre-collapse levels. The paper develops a general framework, proves conditions for existence of a stationary distribution via a Loynes-type construction and PASTA, and specializes to spectrally positive Lévy processes to obtain explicit Laplace-Stieltjes transforms and moment formulas. It provides detailed results for two canonical cases, Reflected Brownian motion with collapse and M/M/1 queue with collapse, expressed through incomplete beta functions, and analyzes heavy-tailed tails for compound Poisson jumps. The findings give a unified approach to growth-collapse phenomena across applied domains, including population genetics, geophysics, cash management, P2P finance, and queueing systems.

Abstract

We consider a Lévy 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 Lévy process is spectrally positive and finally we specialize further to the two cases where the Lévy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.

On reflected Lévy processes with collapse

TL;DR

We study a reflected Lévy process with random, proportional collapses occurring at Poisson times and derive a Lindley-type autoregressive structure for the pre-collapse levels. The paper develops a general framework, proves conditions for existence of a stationary distribution via a Loynes-type construction and PASTA, and specializes to spectrally positive Lévy processes to obtain explicit Laplace-Stieltjes transforms and moment formulas. It provides detailed results for two canonical cases, Reflected Brownian motion with collapse and M/M/1 queue with collapse, expressed through incomplete beta functions, and analyzes heavy-tailed tails for compound Poisson jumps. The findings give a unified approach to growth-collapse phenomena across applied domains, including population genetics, geophysics, cash management, P2P finance, and queueing systems.

Abstract

We consider a Lévy 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 Lévy process is spectrally positive and finally we specialize further to the two cases where the Lévy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.
Paper Structure (9 sections, 3 theorems, 84 equations)

This paper contains 9 sections, 3 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The general model
  3. Lindley-style autoregressive recursions
  4. Stability of the continuous time process
  5. The case where $X$ is spectrally positive
  6. Examples and tail behavior
  7. Reflected Brownian motion with collapse
  8. The M/M/1 queue with collapse
  9. Tail behavior for compound Poisson with negative drift

Key Result

Lemma 1

If $\Pi_n\to0$ as $n\to\infty$ then $z'_n-z_n\to 0$ as $n\to\infty$ for any choice of $z'_0\ge 0$ (smaller, equal or larger than $z_0=v_0$).

Theorems & Definitions (11)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 1 more