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Passage times for a spectrally negative Lévy process with applications to risk theory. (English) Zbl 1076.60038
Let $$X$$ be a Lévy process with a generating triplet $$(\sigma, \nu, a)$$, $$\sigma\geq 0$$, $$a\in\mathbb{R}$$, and $$\nu$$ is supported on $$(-\infty,0)$$. Let $$\tau_x$$ and $$T_x$$ be the first passage times above and below the level $$x$$, and $$l_x$$ and $$T'_x$$ be the last passage times below and above the level $$x$$ after times $$\tau_x$$ and $$T_x$$, respectively. The authors find explicit formulae for the Laplace transforms of $$l_x$$, $$l_x-\tau_x$$ and the joint Laplace transform of $$T_x$$, $$l_x-T_x$$ and $$T'_x-T_x$$. A particular case when jumps constitute a compound Poisson process is considered.

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60E10 Characteristic functions; other transforms
##### Keywords:
first passage time; last passage time
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##### References:
 [1] Asmussen, S. (2000) Ruin Probabilities. Singapore: World Scientific. · Zbl 0960.60003 [2] Avram, F., Kyprianou, A.E. and Pistorius, M.R. (2004) Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab., 14, 215-238. · Zbl 1042.60023 · doi:10.1214/aoap/1075828052 [3] Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press. · Zbl 0861.60003 [4] Bertoin, J. (1997) Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab., 7, 156-169. · Zbl 0880.60077 · doi:10.1214/aoap/1034625257 [5] Bingham, N.H. (1975) Fluctuation theory in continuous time. Adv. Appl. Probab., 7, 705-766. JSTOR: · Zbl 0322.60068 · doi:10.2307/1426397 · links.jstor.org [6] Dickson, D.C.M. and Egídio dos Reis, A.D. (1997) The effect of interest on negative surplus. Insurance Math. Econom., 21, 1-16. · Zbl 0894.90045 · doi:10.1016/S0167-6687(97)00014-0 [7] Doney, C.M. (1991) Hitting probability for spectrally positive Lévy process. J. London Math. Soc., 44, 566-576. · Zbl 0699.60061 · doi:10.1112/jlms/s2-44.3.566 [8] Egídio dos Reis, A.D. (1993) How long is the surplus below zero? Insurance Math. Econom., 12, 23-38. · Zbl 0777.62096 · doi:10.1016/0167-6687(93)90996-3 [9] Egídio dos Reis, A.D. (2000) On the moments of ruin and recovery times. Insurance Math. Econom., 27, 331-343. · Zbl 0988.91044 · doi:10.1016/S0167-6687(00)00056-1 [10] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag. · Zbl 0873.62116 [11] Emery, D.J. (1973) Exit problem for a spectrally positive process. Adv. Appl. Probab., 5, 498-520. JSTOR: · Zbl 0297.60035 · doi:10.2307/1425831 · links.jstor.org [12] Gerber, H.U. (1970) An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuarietidskrift, 53, 205-210. · Zbl 0229.60062 [13] Gerber, H.U. (1990) When does the surplus reach a given target? Insurance Math. Econom., 9, 115-119. · Zbl 0731.62153 · doi:10.1016/0167-6687(90)90022-6 [14] Kyprianou, A.E. and Palmowski, Z. (2005) A martingale review of some fluctuation theory for spectrally negative Lévy processes. In M. É mery, M. Ledoux and M. Yor (eds), Séminaire de Probabilités XXXVIII, Lecture Notes in Math. 1857, pp. 16-29, Berlin: Springer-Verlag. · Zbl 1063.60071 [15] Picard, P. and Lefèvre, C. (1994) On the 1st crossing of the surplus process with a given upper barrier. Insurance Math. Econom., 14, 163-179. · Zbl 0806.62089 · doi:10.1016/0167-6687(94)00010-7 [16] Prabhu, N.U. (1970) Ladder variables for a continuous stochastic process. Z. Wahrscheinlichkeits theorie Verw. Geb., 16, 157-164. · Zbl 0193.45003 · doi:10.1007/BF00535696 [17] Rogers, L.C.G. (1990) The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Probab., 22, 486-487. · Zbl 0698.60063 · doi:10.2307/1427548 [18] Rogers, L.C.G. (2000) Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab., 37, 1173-1180. · Zbl 0981.60048 · doi:10.1239/jap/1014843099 [19] Rogozin, B.A. (1965) On distributions of functionals related to boundary problems for processes with independent increments. Theory Probab. Appl., 11, 580-591. · Zbl 0178.52701 · doi:10.1137/1111062 [20] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. New York: Wiley. · Zbl 0940.60005 [21] Yang, H. and Zhang, L. (2001) Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Probab., 33, 281-291. · Zbl 0978.60104 · doi:10.1239/aap/999187908 [22] Zhang, C. and Wu, R. (2002) Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion. J. Appl. Probab., 39, 517-532. · Zbl 1046.91076 · doi:10.1239/jap/1034082124 [23] Zolotarev, C.M. (1964) The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Theory Probab. Appl., 9, 653-661. · Zbl 0149.12903 · doi:10.1137/1109090
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