Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Philip S. Griffin and Ross A. Maller

Full-text: Open access

Abstract

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the stabilities of the times, $\overline{T} _{b}(r)$ and $T^{*}_{b}(r)$, at which $X$, started with $X_{0}=0$, first leaves the space-time regions $\{(t,y)\in \mathbb{R} ^{2}\colon\ y\le rt^{b},t\ge0\}$ (one-sided exit), or $\{(t,y)\in \mathbb{R} ^{2}\colon\ |y|\le rt^{b},t\ge0\}$ (two-sided exit), $0\le b<1$, 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.

Résumé

Ce papier traite du comportement en temps court d’un processus de Lévy $X$. En particulier, nous étudions la stabilité des temps $\overline{T} _{b}(r)$ et $T^{*}_{b}(r)$ auxquels $X$, partant de $X_{0}=0$, quitte pour la première fois les domaines $\{(t,y)\in \mathbb{R} ^{2}\colon\ y\le rt^{b},t\ge0\}$ (sortie unilatérale), ou $\{(t,y)\in \mathbb{R} ^{2}\colon\ |y|\le rt^{b},t\ge0\}$ (sortie bilatérale), $0\le b<1$, quand $r\downarrow 0$. Nous déterminons si ces temps de passage se comportent ou non comme des fonctions déterministes selon différents modes de convergence : en probabilité, presque sûrement et dans $L^{p}$. Dans de nombreux cas, ceci est équivalent à la stabilité du processus $X$. Le problème analogue à temps grand est aussi discuté.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 208-235.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470132

Digital Object Identifier
doi:10.1214/11-AIHP449

Mathematical Reviews number (MathSciNet)
MR3060154

Zentralblatt MATH identifier
1267.60053

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60F15: Strong theorems 60F25: $L^p$-limit theorems 60K05: Renewal theory

Keywords
Lévy process Passage times across power law boundaries Relative stability Overshoot Random walks

Citation

Griffin, Philip S.; Maller, Ross A. Small and large time stability of the time taken for a Lévy process to cross curved boundaries. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 208--235. doi:10.1214/11-AIHP449. https://projecteuclid.org/euclid.aihp/1359470132


Export citation

References

  • [1] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
  • [2] J. Bertoin, R. A. Doney and R. A. Maller. Passage of Lévy processes across power law boundaries at small times. Ann. Probab. 36 (2008) 160–197.
  • [3] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987.
  • [4] R. M. Blumenthal and R. K. Getoor. Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 (1961) 492–516.
  • [5] R. A. Doney. Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin, 2005.
  • [6] R. A. Doney and P. S. Griffin. Overshoots over curved boundaries. Adv. in Appl. Probab. 35 (2003) 417–448.
  • [7] R. A. Doney and P. S. Griffin. Overshoots over curved boundaries II. Adv. in Appl. Probab. 36 (2004) 1148–1174.
  • [8] R. A. Doney and R. A. Maller. Random walks crossing curved boundaries: Functional limit theorems, stability and asymptotic distributions for exit times and positions. Adv. in Appl. Probab. 32 (2000) 1117–1149.
  • [9] R. A. Doney and R. A. Maller. Stability and attraction to normality for Lévy processes at zero and infinity. J. Theoret. Probab. 15 (2002) 751–792.
  • [10] R. A. Doney and R. A. Maller. Moments of passage times for Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 279–297.
  • [11] R. Durrett. Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomsom Learning, Belmont, 2005.
  • [12] K. B. Erickson. Gaps in the range of nearly increasing processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete 62 (1983) 449–463.
  • [13] P. S. Griffin and R. A. Maller. Stability of the exit time for Lévy processes. Adv. in Appl. Probab. 43 (2011) 712–734.
  • [14] O. Kallenberg. Foundations of Modern Probability. Springer, Berlin, 2001.
  • [15] A. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006.
  • [16] R. A. Maller. Small-time versions of Strassen’s law for Lévy processes. Proc. Lond. Math. Soc. 98 (2009) 531–558.
  • [17] W. E. Pruitt. The growth of random walks and Lévy processes. Ann. Probab. 9 (1981) 948–956.
  • [18] D. O. Siegmund. Some one-sided stopping rules. Ann. Math. Statist. 38 (1967) 1641–1646.