Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of overshoots of Markov additive processes

Leif Döring, Lukas Trottner

Abstract

We prove precise stability results for overshoots of Markov additive processes (MAPs) with finite modulating space. Our approach is based on the Markovian nature of overshoots of MAPs whose mixing and ergodic properties are investigated in terms of the characteristics of the MAP. On our way we extend fluctuation theory of MAPs, contributing among others to the understanding of the Wiener-Hopf factorization for MAPs by generalizing Vigon's équations amicales inversés known for Lévy processes. Using the Lamperti transformation the results can be applied to self-similar Markov processes. Among many possible applications, we study the mixing behavior of stable processes sampled at first hitting times as a concrete example.

Stability of overshoots of Markov additive processes

Abstract

We prove precise stability results for overshoots of Markov additive processes (MAPs) with finite modulating space. Our approach is based on the Markovian nature of overshoots of MAPs whose mixing and ergodic properties are investigated in terms of the characteristics of the MAP. On our way we extend fluctuation theory of MAPs, contributing among others to the understanding of the Wiener-Hopf factorization for MAPs by generalizing Vigon's équations amicales inversés known for Lévy processes. Using the Lamperti transformation the results can be applied to self-similar Markov processes. Among many possible applications, we study the mixing behavior of stable processes sampled at first hitting times as a concrete example.

Paper Structure

This paper contains 10 sections, 29 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and aims of the article
  3. Organization of the paper and main result
  4. Basic notation
  5. Markov additive processes and their fluctuation theory
  6. Stability analysis of overshoots of MAPs
  7. Équations amicales inversés for MAPs
  8. Application to real self-similar Markov processes
  9. Stability of Markov processes
  10. Proof of the resolvent formula

Key Result

Theorem 1

Suppose that the MAP $(\xi,J)$ is upward regular, $J$ is irreducible and the ascending ladder height MAP $(H^+,J^+)$ has a finite first moment. Under mild assumptions on the Lévy system of $(\xi,J)$, $(\mathcal{O}_t,\mathcal{J}_t)_{t \geq 0}$ converges in total variation to a unique stationary distr

Figures (1)

  • Figure 3.1: Path of a compound Poisson subordinator with drift, $\sigma$, and associated overshoot process $\mathcal{O}^\sigma$

Theorems & Definitions (61)

  • Theorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 51 more