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On the future infimum of positive self-similar Markov processes

J. C. Pardo

TL;DR

The paper develops integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes, leveraging the Lamperti representation and time-reversal techniques. By studying the associated last-passage times and their dual processes, it provides a unified framework that extends known LILs for Bessel processes to a broad class of PSSMPs, including regular and log-regular tail regimes and the transient Bessel case. The results yield precise conditions under which the future infimum's envelope grows above given thresholds near 0 and infinity, with explicit asymptotics and examples illustrating the theory. These findings advance understanding of envelope behavior for self-similar Markov processes and offer tools for analyzing related stochastic models via Lévy process representations.

Abstract

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and infinity. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan et al. [11].

On the future infimum of positive self-similar Markov processes

TL;DR

The paper develops integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes, leveraging the Lamperti representation and time-reversal techniques. By studying the associated last-passage times and their dual processes, it provides a unified framework that extends known LILs for Bessel processes to a broad class of PSSMPs, including regular and log-regular tail regimes and the transient Bessel case. The results yield precise conditions under which the future infimum's envelope grows above given thresholds near 0 and infinity, with explicit asymptotics and examples illustrating the theory. These findings advance understanding of envelope behavior for self-similar Markov processes and offer tools for analyzing related stochastic models via Lévy process representations.

Abstract

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and infinity. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan et al. [11].

Paper Structure

This paper contains 9 sections, 25 theorems, 167 equations.

Table of Contents

  1. Introduction and main results
  2. The lower envelope of the last passage times
  3. Proof of Theorems 1 and 2
  4. The regular case
  5. The log regular case
  6. The case when $\xi$ has finite exponential moments
  7. Transient Bessel process.
  8. The upper envelope of increasing self-similar Markov processes.
  9. Examples

Key Result

Lemma 1

Let $\Gamma=X^{(0)}_{U(x)^{-}}$. Then, the process time-reversed at its last passage time below $x>0$, $\hat{X}\stackrel{\hbox{\tiny$($def$)$}}{=}(X^{(0)}_{(U(x)-t)^{-}}, 0\leq t\leq U(x))$ can be described as where $\Gamma$ and $\hat{\xi}$ are independent. Moreover, let $(x_{n})$ be a decreasing sequence which tends to $0$ and such that $x_{1}=x$. If we define $\hat{S}_y=\inf\{t: \hat{X}_{t}\leq

Theorems & Definitions (25)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Corollary 2
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 15 more