The entrance law of the excursion measure of the reflected process for some classes of Lévy processes. (English) Zbl 1472.60082
Summary: We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60G52 | Stable stochastic processes |
| 46N30 | Applications of functional analysis in probability theory and statistics |
Keywords:
Lévy process; supremum process; reflected process; Itô measure; entrance law; stable process; subordinate Brownian motion; integral representationReferences:
| [1] | Baxter, G.; Donsker, M. D., On the distribution of the supremum functional for processes with stationary independent increments, Trans. Am. Math. Soc., 85, 73-87 (1957) · Zbl 0078.32002 |
| [2] | Bertoin, J., Lévy Processes (1996), Melbourne: Cambridge University Press, Melbourne · Zbl 0861.60003 |
| [3] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge: Cambridge University Press, Cambridge · Zbl 0617.26001 |
| [4] | Chaumont, L., On the law of the supremum of Lévy processes, Ann. Probab., 41, 3, 1191-1217 (2013) · Zbl 1277.60081 |
| [5] | Chaumont, L.; Małecki, J., On the asymptotic behavior of the density of the supremum of Lévy processes, Ann. Inst. H. Poincarè Probab. Statist., 52, 3, 1178-1195 (2016) · Zbl 1350.60042 |
| [6] | Chaumont, L.; Małecki, J., Short proofs in extrema of spectrally one sided Lévy processes, Electron. Commun. Probab., 23 (2018) · Zbl 1398.60068 |
| [7] | Chaumont, L., Pellas, T.: Creeping of Lévy processes through deterministic functions. in preparation |
| [8] | Doney, R. A., Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory Held in Saint-Flour, July 6-23, 2005 (2007), Berlin: Springer, Berlin · Zbl 1128.60036 |
| [9] | Hackmann, D.; Kuznetsov, A., A note on the series representation for the density of the supremum of a stable process, Electron. Commun. Probab., 18, 48 (2013) · Zbl 1323.60065 |
| [10] | Hubalek, F.; Kuznetsov, A., A convergent series representation for the density of the supremum of a stable process, Electron. Commun. Probab., 16, 84-95 (2011) · Zbl 1231.60040 |
| [11] | Koyama, S.; Kurokawa, N., Multiple sine functions, Forum Math., 15, 6, 839-876 (2006) · Zbl 1065.11065 |
| [12] | Koyama, S.; Kurokawa, N., Values of the double sine function, J. Number Theory, 123, 1, 204-223 (2007) · Zbl 1160.11044 |
| [13] | Kulczycki, T.; Kwaśnicki, M.; Małecki, J.; Stós, A., Spectral properties of the Cauchy process on half-line and interval, Proc. Lond. Math. Soc., 101, 2, 589-622 (2010) · Zbl 1220.60029 |
| [14] | Kuznetsov, A., On the density of the supremum of a stable process, Stoch. Process. Appl., 123, 983-1003 (2013) · Zbl 1277.60084 |
| [15] | Kuznetsov, A.; Kwaśnicki, M., Spectral analysis of stable processes on the positive half-line, Electron. J. Probab., 23, 10, 1-29 (2018) · Zbl 1390.60173 |
| [16] | Kwaśnicki, M., Spectral analysis of subordinate Brownian motions in half-line, Stud. Math., 206, 3, 21-171 (2011) |
| [17] | Kwaśnicki, M.; Małecki, J.; Ryznar, M., Suprema of Levy processes, Ann. Probab., 41, 3, 2047-2065 (2013) · Zbl 1288.60061 |
| [18] | Kwaśnicki, M.; Małecki, J.; Ryznar, M., First passage times for subordinate Brownian motions, Stoch. Process. Appl., 123, 5, 1820-1850 (2013) · Zbl 1276.60094 |
| [19] | Kyprianou, A., Fluctuations of Lévy Processes with Applications. Introductory Lectures (2014), Heidelberg: Springer, Heidelberg · Zbl 1384.60003 |
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