Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin; Ross A. Maller

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

  • Volume: 49, Issue: 1, page 208-235
  • ISSN: 0246-0203

Abstract

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This paper is concerned with the small time behaviour of a Lévy process X . In particular, we investigate thestabilitiesof the times, T ¯ b ( r ) and T b * ( r ) , at which X , started with X 0 = 0 , first leaves the space-time regions { ( t , y ) ∈ ℝ 2 : y ≤ r t b , t ≥ 0 } (one-sided exit), or { ( t , y ) ∈ ℝ 2 : | y | ≤ r t b , t ≥ 0 } (two-sided exit), 0 ≤ b l t ; 1 , as r ↓ 0 . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L p . In many instances these are seen to be equivalent to relative stability of the process X itself. The analogous large time problem is also discussed.

How to cite

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Griffin, Philip S., and Maller, Ross A.. "Small and large time stability of the time taken for a Lévy process to cross curved boundaries." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 208-235. <http://eudml.org/doc/272078>.

@article{Griffin2013,
abstract = {This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate thestabilitiesof the times, $\overline\{T\} _\{b\}(r)$ and $T^\{*\}_\{b\}(r)$, at which $X$, started with $X_\{0\}=0$, first leaves the space-time regions $\lbrace (t,y)\in \mathbb \{R\} ^\{2\}\colon \ y\le rt^\{b\},t\ge 0\rbrace $ (one-sided exit), or $\lbrace (t,y)\in \mathbb \{R\} ^\{2\}\colon \ |y|\le rt^\{b\},t\ge 0\rbrace $ (two-sided exit), $0\le b&lt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^\{p\}$. In many instances these are seen to be equivalent to relative stability of the process $X$ itself. The analogous large time problem is also discussed.},
author = {Griffin, Philip S., Maller, Ross A.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy process; passage times across power law boundaries; relative stability; overshoot; random walks; Levy process; passage times across power law boundary; overshoot random walk},
language = {eng},
number = {1},
pages = {208-235},
publisher = {Gauthier-Villars},
title = {Small and large time stability of the time taken for a Lévy process to cross curved boundaries},
url = {http://eudml.org/doc/272078},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Griffin, Philip S.
AU - Maller, Ross A.
TI - Small and large time stability of the time taken for a Lévy process to cross curved boundaries
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 208
EP - 235
AB - This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate thestabilitiesof the times, $\overline{T} _{b}(r)$ and $T^{*}_{b}(r)$, at which $X$, started with $X_{0}=0$, first leaves the space-time regions $\lbrace (t,y)\in \mathbb {R} ^{2}\colon \ y\le rt^{b},t\ge 0\rbrace $ (one-sided exit), or $\lbrace (t,y)\in \mathbb {R} ^{2}\colon \ |y|\le rt^{b},t\ge 0\rbrace $ (two-sided exit), $0\le b&lt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^{p}$. In many instances these are seen to be equivalent to relative stability of the process $X$ itself. The analogous large time problem is also discussed.
LA - eng
KW - Lévy process; passage times across power law boundaries; relative stability; overshoot; random walks; Levy process; passage times across power law boundary; overshoot random walk
UR - http://eudml.org/doc/272078
ER -

References

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