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, Volume 49 (2013) no. 1, p. 208-235

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the stabilities of the times, ${\overline{T}}_{b}\left(r\right)$ and ${T}_{b}^{*}\left(r\right)$, at which $X$, started with ${X}_{0}=0$, first leaves the space-time regions $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}y\le r{t}^{b},t\ge 0\right\}$ (one-sided exit), or $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}|y|\le r{t}^{b},t\ge 0\right\}$ (two-sided exit), $0\le b<1$, 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.

Ce papier traite du comportement en temps court d’un processus de Lévy $X$. En particulier, nous étudions la stabilité des temps ${\overline{T}}_{b}\left(r\right)$ et ${T}_{b}^{*}\left(r\right)$ auxquels $X$, partant de ${X}_{0}=0$, quitte pour la première fois les domaines $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}y\le r{t}^{b},t\ge 0\right\}$ (sortie unilatérale), ou $\left\{\left(t,y\right)\in {ℝ}^{2}:\phantom{\rule{4pt}{0ex}}|y|\le r{t}^{b},t\ge 0\right\}$ (sortie bilatérale), $0\le b<1$, quand $r↓0$. 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.

DOI : https://doi.org/10.1214/11-AIHP449
Classification:  60G51,  60F15,  60F25,  60K05
Keywords: Lévy process, passage times across power law boundaries, relative stability, overshoot, random walks
@article{AIHPB_2013__49_1_208_0,
author = {Griffin, Philip S. and Maller, Ross A.},
title = {Small and large time stability of the time taken for a L\'evy process to cross curved boundaries},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {49},
number = {1},
year = {2013},
pages = {208-235},
doi = {10.1214/11-AIHP449},
zbl = {1267.60053},
mrnumber = {3060154},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2013__49_1_208_0}
}

Griffin, Philip S.; 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, Volume 49 (2013) no. 1, pp. 208-235. doi : 10.1214/11-AIHP449. http://www.numdam.org/item/AIHPB_2013__49_1_208_0/

[1] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0938.60005

[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. | MR 2370602 | Zbl 1140.60025

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR 898871 | Zbl 0667.26003

[4] R. M. Blumenthal and R. K. Getoor. Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 (1961) 492-516. | MR 123362 | Zbl 0097.33703

[5] R. A. Doney. Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin, 2005. | Zbl 1128.60036

[6] R. A. Doney and P. S. Griffin. Overshoots over curved boundaries. Adv. in Appl. Probab. 35 (2003) 417-448. | MR 1970482 | Zbl 1037.60043

[7] R. A. Doney and P. S. Griffin. Overshoots over curved boundaries II. Adv. in Appl. Probab. 36 (2004) 1148-1174. | MR 2119858 | Zbl 1064.60091

[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. | MR 1808917 | Zbl 0976.60082

[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. | MR 1922446 | Zbl 1015.60043

[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. | Numdam | MR 2060454 | Zbl 1042.60025

[11] R. Durrett. Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomsom Learning, Belmont, 2005. | MR 1068527 | Zbl 0709.60002

[12] K. B. Erickson. Gaps in the range of nearly increasing processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete 62 (1983) 449-463. | MR 690570 | Zbl 0488.60080

[13] P. S. Griffin and R. A. Maller. Stability of the exit time for Lévy processes. Adv. in Appl. Probab. 43 (2011) 712-734. | MR 2858218 | Zbl 1232.60037

[14] O. Kallenberg. Foundations of Modern Probability. Springer, Berlin, 2001. | MR 1876169 | Zbl 0892.60001

[15] A. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. | MR 2250061 | Zbl pre06176054

[16] R. A. Maller. Small-time versions of Strassen's law for Lévy processes. Proc. Lond. Math. Soc. 98 (2009) 531-558. | MR 2481958 | Zbl 1157.60044

[17] W. E. Pruitt. The growth of random walks and Lévy processes. Ann. Probab. 9 (1981) 948-956. | MR 632968 | Zbl 0477.60033

[18] D. O. Siegmund. Some one-sided stopping rules. Ann. Math. Statist. 38 (1967) 1641-1646. | MR 220402 | Zbl 0183.20707