Stability of the overshoot for Lévy processes
type: Item
instance of: scholarly article
title: Stability of the overshoot for Lévy processes
zbMATH Open document ID: 1016.60052
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \(X_t\) be a real-valued Lévy process (i.e., a stochastically continuous random process with stationary and independent increments) with characteristic exponent \(\psi\) defined by \(\mathbb{E}e^{i\xi X_t} = e^{-t\psi(\xi)}\). Further, let \((\gamma,\sigma^2,\Pi)\) be the Lévy triplet which uniquely characterises \(\psi\) via the Lévy-Khinchin formula. The authors study the behaviour of the overshoot of the process upon exiting a strip resp. a halfspace. Denote by \[ T(r) = \inf\{t>0: |X_t|> r\} \quad\text{resp.}\quad T^*(r) = \inf\{t>0: X_t > r\} \] the first exit time of \(X_t\) from a symmetric strip resp.\ a halfplane. The authors etablish necessary and sufficient criteria for the following (almost sure or in probability) limits \[ \lim_{r\downarrow 0} \frac{|X(T(r))|}{r} = 1 \quad\text{and}\quad \lim_{r\downarrow 0} \frac{X(T^*(r))}{r} = 1 \tag{\(*\)} \] to happen. Similar limits for the two-sided exit problem are valid also when \(r\to\infty\), the proofs for this case are only sketched and resemble in scope and method the results for random walks given in the paper by \textit{P. S. Griffin} and \textit{R. A. Maller} [Adv. Appl. Probab. 30, 181-196 (1998; Zbl 0905.60064)]. It is shown that the first limit in \((*)\) holds in probability if and only if either (a) \(X_t\) lies in the domain of attraction of the normal law, that is, if \(X_t/b(t) \to N(0,1)\) in law as \(t\downarrow 0\) for some deterministic \(b(t)\) or if (b) \(X_t\) is relatively stable, i.e., if \(X_t/b(t) \to \pm 1\) in probability as \(t\downarrow 0\) for some deterministic \(b(t)\). (This explains the word stability in the title of the paper.) The criteria for both limits in \((*)\) to take place almost surely are given in terms of (functions of) the Lévy triple, using in particular certain equivalents of truncated mean and variance of the process and (one-sided) tails of the Lévy measure \(\Pi\). The proofs use two main ingredients: formulae for the distribution that the process cross…
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1906049
zbMATH Keywords: Lévy processes
zbMATH Keywords: Lévy characteristics
zbMATH Keywords: independent increments
zbMATH Keywords: exit systems
zbMATH Keywords: exit times
zbMATH Keywords: first passage times
zbMATH Keywords: local behaviour
author: Ronald A. Doney
author: Ross A. Maller
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1020107765
OpenAlex ID: W2057764228
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DOI: 10.1214/AOP/1020107765
description: scientific article; zbMATH DE number 1906049