Bernoulli

  • Bernoulli
  • Volume 11, Number 3 (2005), 511-522.

Passage times for a spectrally negative Lévy process with applications to risk theory

Sung Nok Chiu and Chuancun Yin

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Abstract

The distributions of the last passage time at a given level and the joint distributions of the last passage time, the first passage time and their difference for a general spectrally negative process are derived in the form of Laplace transforms. The results are applied to risk theory.

Article information

Source
Bernoulli, Volume 11, Number 3 (2005), 511-522.

Dates
First available in Project Euclid: 5 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.bj/1120591186

Digital Object Identifier
doi:10.3150/bj/1120591186

Mathematical Reviews number (MathSciNet)
MR2146892

Zentralblatt MATH identifier
1076.60038

Keywords
first passage time last passage time spectrally negative Lévy process risk theory

Citation

Nok Chiu, Sung; Yin, Chuancun. Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11 (2005), no. 3, 511--522. doi:10.3150/bj/1120591186. https://projecteuclid.org/euclid.bj/1120591186


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References

  • [1] Asmussen, S. (2000) Ruin Probabilities. Singapore: World Scientific.
  • [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.
  • [3] Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press.
  • [4] Bertoin, J. (1997) Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab., 7, 156-169.
  • [5] Bingham, N.H. (1975) Fluctuation theory in continuous time. Adv. Appl. Probab., 7, 705-766.
  • [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.
  • [7] Doney, C.M. (1991) Hitting probability for spectrally positive Lévy process. J. London Math. Soc., 44, 566-576.
  • [8] Egídio dos Reis, A.D. (1993) How long is the surplus below zero? Insurance Math. Econom., 12, 23-38.
  • [9] Egídio dos Reis, A.D. (2000) On the moments of ruin and recovery times. Insurance Math. Econom., 27, 331-343.
  • [10] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.
  • [11] Emery, D.J. (1973) Exit problem for a spectrally positive process. Adv. Appl. Probab., 5, 498-520.
  • [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.
  • [13] Gerber, H.U. (1990) When does the surplus reach a given target? Insurance Math. Econom., 9, 115-119.
  • [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.
  • [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.
  • [16] Prabhu, N.U. (1970) Ladder variables for a continuous stochastic process. Z. Wahrscheinlichkeits theorie Verw. Geb., 16, 157-164.
  • [17] Rogers, L.C.G. (1990) The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Probab., 22, 486-487.
  • [18] Rogers, L.C.G. (2000) Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab., 37, 1173-1180.
  • [19] Rogozin, B.A. (1965) On distributions of functionals related to boundary problems for processes with independent increments. Theory Probab. Appl., 11, 580-591.
  • [20] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. New York: Wiley.
  • [21] Yang, H. and Zhang, L. (2001) Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Probab., 33, 281-291.
  • [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.
  • [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.