This paper is concerned with the small time behaviour of a Lévy process . In particular, we investigate the stabilities of the times, and , at which , started with , first leaves the space-time regions (one-sided exit), or (two-sided exit), , as . 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 . In many instances these are seen to be equivalent to relative stability of the process itself. The analogous large time problem is also discussed.
Ce papier traite du comportement en temps court d’un processus de Lévy . En particulier, nous étudions la stabilité des temps et auxquels , partant de , quitte pour la première fois les domaines (sortie unilatérale), ou (sortie bilatérale), , quand . 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 . Dans de nombreux cas, ceci est équivalent à la stabilité du processus . Le problème analogue à temps grand est aussi discuté.
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}, pages = {208--235}, publisher = {Gauthier-Villars}, volume = {49}, number = {1}, year = {2013}, doi = {10.1214/11-AIHP449}, mrnumber = {3060154}, zbl = {1267.60053}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP449/} }
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 SP - 208 EP - 235 VL - 49 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP449/ DO - 10.1214/11-AIHP449 LA - en ID - AIHPB_2013__49_1_208_0 ER -
%0 Journal Article %A Griffin, Philip S. %A Maller, Ross A. %T Small and large time stability of the time taken for a Lévy process to cross curved boundaries %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 208-235 %V 49 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP449/ %R 10.1214/11-AIHP449 %G en %F 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/articles/10.1214/11-AIHP449/
[1] Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
.[2] Passage of Lévy processes across power law boundaries at small times. Ann. Probab. 36 (2008) 160-197. | MR | Zbl
, and .[3] Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl
, and .[4] Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 (1961) 492-516. | MR | Zbl
and .[5] Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin, 2005. | Zbl
.[6] Overshoots over curved boundaries. Adv. in Appl. Probab. 35 (2003) 417-448. | MR | Zbl
and .[7] Overshoots over curved boundaries II. Adv. in Appl. Probab. 36 (2004) 1148-1174. | MR | Zbl
and .[8] 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
and .[9] Stability and attraction to normality for Lévy processes at zero and infinity. J. Theoret. Probab. 15 (2002) 751-792. | MR | Zbl
and .[10] Moments of passage times for Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 279-297. | Numdam | MR | Zbl
and .[11] Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomsom Learning, Belmont, 2005. | MR | Zbl
.[12] Gaps in the range of nearly increasing processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete 62 (1983) 449-463. | MR | Zbl
.[13] Stability of the exit time for Lévy processes. Adv. in Appl. Probab. 43 (2011) 712-734. | MR | Zbl
and .[14] Foundations of Modern Probability. Springer, Berlin, 2001. | MR | Zbl
.[15] Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. | MR
.[16] Small-time versions of Strassen's law for Lévy processes. Proc. Lond. Math. Soc. 98 (2009) 531-558. | MR | Zbl
.[17] The growth of random walks and Lévy processes. Ann. Probab. 9 (1981) 948-956. | MR | Zbl
.[18] Some one-sided stopping rules. Ann. Math. Statist. 38 (1967) 1641-1646. | MR | Zbl
.Cited by Sources: