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

Griffin, Philip S.; Maller, Ross A.

Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, pp. 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 $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (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.

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 $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (sortie unilatérale), ou $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (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.

Classification: 60G51, 60F15, 60F25, 60K05
Keywords: Lévy process, passage times across power law boundaries, relative stability, overshoot, random walks

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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