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The structure of entrance and exit at infinity for time-changed Lévy processes

Samuel Baguley, Leif Döring, Quan Shi

Abstract

Studying the behaviour of Markov processes at boundary points of the state space has a long history, dating back all the way to William Feller. With different motivations in mind entrance and exit questions have been explored for different discontinuous Markov processes in the past two decades. Proofs often use time-change techniques and rely on problem specific knowledge such branching or scaling properties. In this article we ask how far techniques can be pushed with as little as possible model assumptions. We give sharp conditions on time-changed Lévy processes to allow entrance and regular boundary point at infinity. The main tool we introduce is a generalised scaling property that holds for all time-changed Lévy processes and can be used to extend scaling arguments for self-similar Markov processes.

The structure of entrance and exit at infinity for time-changed Lévy processes

Abstract

Studying the behaviour of Markov processes at boundary points of the state space has a long history, dating back all the way to William Feller. With different motivations in mind entrance and exit questions have been explored for different discontinuous Markov processes in the past two decades. Proofs often use time-change techniques and rely on problem specific knowledge such branching or scaling properties. In this article we ask how far techniques can be pushed with as little as possible model assumptions. We give sharp conditions on time-changed Lévy processes to allow entrance and regular boundary point at infinity. The main tool we introduce is a generalised scaling property that holds for all time-changed Lévy processes and can be used to extend scaling arguments for self-similar Markov processes.

Paper Structure

This paper contains 20 sections, 37 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Main results
  4. Entrance at infinity
  5. Techniques used for the proofs
  6. Examples and connections with previous results
  7. Time-changed Lévy processes and the R-scaling invariance property
  8. Preliminaries: fluctuation theory of Lévy processes
  9. Explosion time
  10. Continuity at the initial state
  11. A new invariance property for time-changed Lévy processes
  12. Coming down from infinity
  13. Sufficient conditions
  14. Necessary conditions
  15. The speed of coming down from infinity
  16. ...and 5 more sections

Key Result

Theorem 2.4

The time-changed Lévy process has $+\infty$ as an entrance boundary if and only if

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: $+\infty$ is an entrance boundary
  • Definition 2.5: $R$-scaling invariance
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Theorem 2.10: $+\infty$ is a regular boundary
  • ...and 81 more