×

On the law of the supremum of Lévy processes. (English) Zbl 1277.60081

Summary: We show that the law of the overall supremum \(\overline{X}_{t}=\sup_{s\leq t}X_{s}\) of a Lévy process \(X\), before the deterministic time \(t\) is equivalent to the average occupation measure \(\mu_{t}^{+}(dx)=\int_{0}^{t}\operatorname{P} (X_{s}\in dx)\,ds\), whenever 0 is regular for both open halflines \((-\infty,0)\) and \((0,\infty)\). In this case, \(\operatorname{P} (\overline{X}_{t}\in dx)\) is absolutely continuous for some (and hence for all) \(t>0\) if and only if the resolvent measure of \(X\) is absolutely continuous. We also study the cases where 0 is not regular for both halflines. Then, we give absolute continuity criteria for the laws of \((g_{t},\overline{X}_{t})\) and \((g_{t},\overline{X}_{t},X_{t})\), where \(g_{t}\) is the time at which the supremum occurs before \(t\). The proofs of these results use an expression of the joint law \(\operatorname{P} (g_{t}\in ds,X_{t}\in dx,\overline{X}_{t}\in dy)\) in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made explicit (partly) in some particular instances.

MSC:

60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes

References:

[1] Alili, L. and Chaumont, L. (2001). A new fluctuation identity for Lévy processes and some applications. Bernoulli 7 557-569. · Zbl 1003.60045 · doi:10.2307/3318502
[2] Alili, L., Chaumont, L. and Doney, R. A. (2005). On a fluctuation identity for random walks and Lévy processes. Bull. Lond. Math. Soc. 37 141-148. · Zbl 1063.60062 · doi:10.1112/S0024609304003789
[3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[4] Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 273-296. · Zbl 0238.60036 · doi:10.1007/BF00534892
[5] Bouleau, N. and Denis, L. (2009). Energy image density property and the lent particle method for Poisson measures. J. Funct. Anal. 257 1144-1174. · Zbl 1174.60023 · doi:10.1016/j.jfa.2009.03.004
[6] Chaumont, L. and Małecki, J. (2012). Density of the supremum and entrance law of the reflected excursion of Lévy processes. Work in progress.
[7] Cordero, F. (2010). Sur la théorie des excursions pour des processus de Lévy symétriques stables d’indice \(\alpha\in\,]1,2]\), et quelques applications. Ph.D. thesis, Univ. Paris 6.
[8] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897 . Springer, Berlin. · Zbl 1128.60036
[9] Doney, R. A. and Rivero, V. (2011). Asymptotic behaviour of first passage time distributions for Lévy processes. Preprint. Available at . 1107.4415v1 · Zbl 1286.60042
[10] Doney, R. A. and Savov, M. S. (2010). The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 316-326. · Zbl 1185.60052 · doi:10.1214/09-AOP479
[11] Fukushima, M. (1976). Potential theory of symmetric Markov processes and its applications. In Proceedings of the Third Japan-USSR Symposium on Probability Theory ( Tashkent , 1975). Lecture Notes in Math. 550 119-133. Springer, Berlin. · Zbl 0349.60077 · doi:10.1007/BFb0077487
[12] Kallenberg, O. (1981). Splitting at backward times in regenerative sets. Ann. Probab. 9 781-799. · Zbl 0526.60061 · doi:10.1214/aop/1176994308
[13] Kuznetsov, A. (2011). On extrema of stable processes. Ann. Probab. 39 1027-1060. · Zbl 1218.60037 · doi:10.1214/10-AOP577
[14] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[15] Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien . Gauthier-Villars & Cie, Paris. · Zbl 0137.11602
[16] Orey, S. (1968). On continuity properties of infinitely divisible distribution functions. Ann. Math. Statist. 39 936-937. · Zbl 0172.22101 · doi:10.1214/aoms/1177698325
[17] Pečerskiĭ, E. A. and Rogozin, B. A. (1969). The combined distributions of the random variables connected with the fluctuations of a process with independent increments. Teor. Verojatnost. i Primenen. 14 431-444. · Zbl 0194.49001
[18] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68 . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001
[19] Yano, Y. (2010). A remarkable \(\sigma\)-finite measure unifying supremum penalisations for a stable Lévy process. · Zbl 1282.60051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.