×

zbMATH — the first resource for mathematics

Occupation times of spectrally negative Lévy processes with applications. (English) Zbl 1227.60061
Summary: We compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative Lévy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative Lévy process and its Laplace exponent. Applications to insurance risk models are also presented.

MSC:
60G51 Processes with independent increments; Lévy processes
60E10 Characteristic functions; other transforms
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bertoin, J., Lévy processes, (1996), Cambridge University Press · Zbl 0861.60003
[2] Bertoin, J., Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval, Ann. appl. probab., 7, 1, 156-169, (1997) · Zbl 0880.60077
[3] Biard, R.; Loisel, S.; Macci, C.; Veraverbeke, N., Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation, J. math. anal. appl., 367, 2, 535-549, (2010) · Zbl 1192.91111
[4] T. Chan, A.E. Kyprianou, M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probability Theory and Related Fields (in press). · Zbl 1259.60050
[5] Chesney, M.; Jeanblanc-Picqué, M.; Yor Brownian, M., Excursions and Parisian barrier options, Adv. appl. probab., 29, 1, 165-184, (1997) · Zbl 0882.60042
[6] I. Czarna, Z. Palmowski, Ruin probability with Parisian delay for a spectrally negative Lévy risk process, 2010, arXiv:1003.4299v1 [math.PR]. · Zbl 1232.60036
[7] A. Dassios, S. Wu, Parisian ruin with exponential claims, preprint, 2009.
[8] dos Reis, A.E., How long is the surplus below zero?, Insurance math. econom., 12, 1, 23-38, (1993) · Zbl 0777.62096
[9] Hubalek, F.; Kyprianou, A.E., Old and new examples of scale functions for spectrally negative Lévy processes, () · Zbl 1274.60148
[10] Itô, K.; McKean, H.P., Diffusion processes and their sample paths, (1974), Springer-Verlag Berlin · Zbl 0285.60063
[11] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer-Verlag New York · Zbl 0734.60060
[12] Klüppelberg, C.; Kyprianou, A.E., On extreme ruinous behaviour of Lévy insurance risk processes, J. appl. probab., 43, 594-598, (2006) · Zbl 1118.60071
[13] A. Kuznetsov, A.E. Kyprianou, V. Rivero, The theory of scale functions for spectrally negative Lévy processes, 2011, arXiv:1104.1280v1 [math.PR]. · Zbl 1261.60047
[14] Kyprianou, A.E., Introductory lectures on fluctuations of Lévy processes with applications, (2006), Universitext. Springer-Verlag Berlin · Zbl 1104.60001
[15] Kyprianou, A.E.; Patie, P., A Ciesielski-Taylor type identity for positive self-similar Markov processes, Ann. inst. H. Poincaré, 47, 3, 917-928, (2011) · Zbl 1231.60031
[16] Kyprianou, A.E.; Rivero, V., Special, conjugate and complete scale functions for spectrally negative Lévy processes, Electron. J. probab., 13, 1672-1701, (2008) · Zbl 1193.60064
[17] D. Landriault, J.-F. Renaud, X. Zhou, Insurance risk models with Parisian implementation delays, 2010, ssrn.com/abstract=1744193. · Zbl 1319.60098
[18] R.L. Loeffen, I. Czarna, Z. Palmowski, Parisian ruin probability for spectrally negative Lévy processes, 2011, arXiv:1102.4055v1 [math.PR]. · Zbl 1267.60054
[19] Loeffen, R.L.; Renaud, J.-F., De finetti’s optimal dividends problem with an affine penalty function at ruin, Insurance math. econom., 46, 1, 98-108, (2010) · Zbl 1231.91212
[20] Obłój, J.; Pistorius, M., On an explicit Skorokhod embedding for spectrally negative Lévy processes, J. theoret. probab., 22, 2, 418-440, (2009) · Zbl 1166.60028
[21] Surya, B.A., Evaluating scale functions of spectrally negative Lévy processes, J. appl. probab., 45, 1, 135-149, (2008) · Zbl 1140.60027
[22] C. Yin, K. Yuen, Some exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory, 2011, arXiv:1101.0445v2 [math.PR].
[23] Zhang, C.; Wu, R., Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion, J. appl. probab., 39, 3, 517-532, (2002) · Zbl 1046.91076
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.