Annals of Applied Probability

On the optimal dividend problem for a spectrally negative Lévy process

Florin Avram, Zbigniew Palmowski, and Martijn R. Pistorius

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In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

Article information

Ann. Appl. Probab., Volume 17, Number 1 (2007), 156-180.

First available in Project Euclid: 13 February 2007

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Zentralblatt MATH identifier

Primary: 60J99: None of the above, but in this section
Secondary: 93E20: Optimal stochastic control 60G51: Processes with independent increments; Lévy processes

Lévy process dividend problem local time reflection scale function fluctuation theory


Avram, Florin; Palmowski, Zbigniew; Pistorius, Martijn R. On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17 (2007), no. 1, 156--180. doi:10.1214/105051606000000709.

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  • Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 261–308.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
  • 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.
  • 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.
  • Bertoin, J. (1995). On the first exit time of a completely asymmetric Lévy process from a finite interval. Bull. London Math. Soc. 28 514–520.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 156–169.
  • Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705–766.
  • Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov process. Z. Wahrsch. Verw. Gebiete 54 161–219.
  • De Finetti, B. (1957). Su un'impostazione alternativa dell teoria colletiva del rischio. Trans. XV Intern. Congress Act. 2 433–443.
  • Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. Hübner Foundation for Insurance Education, Philadelphia.
  • Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: Analysis with Brownian motion. North American Actuarial J. 8 1–20.
  • Harrison, J. M. and Taylor, A. J. (1978). Optimal control of a Brownian storage system. Stochastic Process. Appl. 6 179–194.
  • Irbäck, J. (2003). Asymptotic theory for a risk process with a high dividend barrier. Scand. Actuarial J. 2 97–118.
  • Jeanblanc, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50 257–277.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 16–29. Springer, Berlin.
  • Lambert, A. (2000). Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. H. Poincaré Probab. Statist. 36 251–274.
  • Løkka, A. and Zervos, M. (2005). Optimal dividend and issuance of equity policies in the presence of proportional costs. Preprint.
  • Pistorius, M. R. (2003). On doubly reflected completely asymmetric Lévy processes. Stochastic Process. Appl. 107 131–143.
  • Pistorius, M. R. (2004). On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum. J. Theoret. Probab. 17 183–220.
  • Pistorius, M. R. (2006). An excursion theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. Sém. Probab. To appear.
  • Protter, P. (1995). Stochastic Integration and Differential Equations. Springer, Berlin.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Zhou, X. (2005). On a classical risk model with a constant dividend barrier. North American Actuarial J. 9 1–14.