Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada; Steffen Dereich

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 1, page 236-251
  • ISSN: 0246-0203

Abstract

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We consider the one-sided exit problem – also called one-sided barrier problem – for ( α -fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function α ↦ θ ( α ) such that the probability that any α -fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time T behaves as T - θ ( α ) + o ( 1 ) for large T . We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary). This, in particular, extends Sinai’s result on the survival exponent θ ( 1 ) = 1 / 4 for the integrated simple random walk to general random walks with some finite exponential moment.

How to cite

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Aurzada, Frank, and Dereich, Steffen. "Universality of the asymptotics of the one-sided exit problem for integrated processes." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 236-251. <http://eudml.org/doc/272048>.

@article{Aurzada2013,
abstract = {We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha $-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta (\alpha )$ such that the probability that any $\alpha $-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^\{-\theta (\alpha )+\mathrm \{o\}(1)\}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary). This, in particular, extends Sinai’s result on the survival exponent $\theta (1)=1/4$ for the integrated simple random walk to general random walks with some finite exponential moment.},
author = {Aurzada, Frank, Dereich, Steffen},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {integrated brownian motion; integrated Lévy process; integrated random walk; lower tail probability; moving boundary; one-sided barrier problem; one-sided exit problem; persistence probabilities; survival exponent; fractionally integrated Lévy process; moving barrier; persistence exponent; random polynomials},
language = {eng},
number = {1},
pages = {236-251},
publisher = {Gauthier-Villars},
title = {Universality of the asymptotics of the one-sided exit problem for integrated processes},
url = {http://eudml.org/doc/272048},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Aurzada, Frank
AU - Dereich, Steffen
TI - Universality of the asymptotics of the one-sided exit problem for integrated processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 236
EP - 251
AB - We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha $-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta (\alpha )$ such that the probability that any $\alpha $-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta (\alpha )+\mathrm {o}(1)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary). This, in particular, extends Sinai’s result on the survival exponent $\theta (1)=1/4$ for the integrated simple random walk to general random walks with some finite exponential moment.
LA - eng
KW - integrated brownian motion; integrated Lévy process; integrated random walk; lower tail probability; moving boundary; one-sided barrier problem; one-sided exit problem; persistence probabilities; survival exponent; fractionally integrated Lévy process; moving barrier; persistence exponent; random polynomials
UR - http://eudml.org/doc/272048
ER -

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