Journal of Applied Probability

The moments of the discounted loss and the discounted dividends for a spectrally negative Lévy risk process

Esther Frostig

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Consider a spectrally negative risk process where, on ruin, the deficit is immediately paid, and the process restarts from 0. When the process reaches a threshold b, all the surplus above b is paid as dividend. Applying the theory of exit times for a spectrally negative Lévy process and its reflection at the maximum and at the minimum, we obtain recursive formulae for the following moments. (i) The moments of the discounted loss until the process reaches b. This is equivalent to the moments of the discounted dividends in the dual model under the barrier strategy. (ii) The moments of the discounted loss for models with and without a dividend barrier for the infinite horizon. (iii) The moments of the discounted dividends for the infinite horizon.

Article information

Source
J. Appl. Probab., Volume 52, Number 3 (2015), 665-687.

Dates
First available in Project Euclid: 22 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1445543839

Digital Object Identifier
doi:10.1239/jap/1445543839

Mathematical Reviews number (MathSciNet)
MR3414984

Zentralblatt MATH identifier
1326.60063

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 91B30: Risk theory, insurance

Keywords
Dividends barrier strategy capital injection exit times reflected process scale function dual model

Citation

Frostig, Esther. The moments of the discounted loss and the discounted dividends for a spectrally negative Lévy risk process. J. Appl. Probab. 52 (2015), no. 3, 665--687. doi:10.1239/jap/1445543839. https://projecteuclid.org/euclid.jap/1445543839


Export citation

References

  • Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.
  • Asmussen, S., Højgaard, B. and Taksar, M. (2000). Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Finance Stoch. 4, 299–324.
  • Avanzi, B. and Gerber, H. U. (2008) Optimal dividends in the dual model with diffusion. ASTIN Bull. 38, 653–667.
  • Avanzi, B., Gerber, H. U. and Shiu, E. W. S. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111–123.
  • Avanzi, B., Shen, J. and Wong, B. (2011). Optimal dividends and capital injections in the dual model with diffusion. ASTIN Bull. 41, 611–644.
  • Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.
  • Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359–372.
  • Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
  • Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.
  • Cheung, E. C. K. and Drekic, S. (2008). Dividend moments in the dual risk model: exact and approximate approaches. ASTIN Bull. 38, 399–422.
  • Dickson, D. C. M. and Waters, H. R. (2004). Some optimal dividends problems. ASTIN Bull. 34, 49–74.
  • De Finetti, B. (1957). Su un'impostazione alternativa dell teoria collettiva del rischio. Trans. XVth Internat. Congr. Actuaries 2, 433–443.
  • Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Schweiz. Verein. Versicherungsmath. 69, 185–228.
  • Hubalek, F. and Kyprianou, A. E. (2011). Old and new examples of scale functions for spectrally negative Lévy processes. In Seminar on Stochastic Analysis, Random Fields and Applications VI (Progress Prob. 63), Birkhäuser, Basel, pp. 119–145.
  • Kulenko, N. and Schmidli, H. (2008). Optimal dividend strategies on a Cramér–Lundberg model with capital injections. Insurance Math. Econom. 43, 270–278.
  • Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428–443.
  • Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669–1680.
  • Mijatović, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812–3836.
  • Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.
  • Renaud, J-F. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420–427.
  • Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.
  • Suprun, V. N. (1976). Problem of destruction and resolvent of a terminating process with independent increments. Ukrainian Math. J. 28, 39–51.