The Annals of Applied Probability

On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes

R. L. Loeffen

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We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433–443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156–180] studied the case when the risk process is modeled by a general spectrally negative Lévy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.

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Ann. Appl. Probab. Volume 18, Number 5 (2008), 1669-1680.

First available in Project Euclid: 30 October 2008

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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 stochastic control dividend problem scale function complete monotonicity


Loeffen, R. L. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18 (2008), no. 5, 1669--1680. doi:10.1214/07-AAP504.

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  • [1] Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability 2. World Scientific, River Edge, NJ.
  • [2] Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20 1–15.
  • [3] Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17 156–180.
  • [4] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 261–308.
  • [5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [6] Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Die Grundlehren der mathematischen Wissenschaften 172. Springer, New York.
  • [7] Chan, T. and Kyprianou, A. E. (2009). Smoothness of scale functions for spectrally negative Lévy processes. Submitted.
  • [8] de Finetti, B. (1957). Su un’impostazion alternativa dell teoria collecttiva del rischio. In Trans. XVth Internat. Congress Actuaries 2 433–443.
  • [9] Dickson, D. C. M. and Waters, H. R. (2004). Some optimal dividends problems. Astin Bull. 34 49–74.
  • [10] Dufresne, F. and Gerber, H. U. (1993). The probability of ruin for the inverse Gaussian and related processes. Insurance Math. Econom. 12 9–22.
  • [11] Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. Astin Bull. 21 177–192.
  • [12] Frostig, E. (2005). The expected time to ruin in a risk process with constant barrier via martingales. Insurance Math. Econom. 37 216–228.
  • [13] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuar. J. 1 59–74.
  • [14] Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mitt. Ver. Schweiz. Versich. Math. 69 185–227.
  • [15] Gerber, H. U., Lin, X. S. and Yang, H. (2006). A note on the dividends-penalty identity and the optimal dividend barrier. Astin Bull. 36 489–503.
  • [16] Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: Analysis with Brownian motion. N. Am. Actuar. J. 8 1–20.
  • [17] Hubalek, F. and Kyprianou, A. E. (2007). Old and new examples of scale functions for spectrally negative Lévy processes. Preprint.
  • [18] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14 1378–1397.
  • [19] Irbäck, J. (2003). Asymptotic theory for a risk process with a high dividend barrier. Scand. Actuar. J. 2 97–118.
  • [20] Jacob, N. (2001). Pseudo Differential Operators and Markov Processes, Vol. I: Fourier Analysis and Semigroups. Imperial College Press, London.
  • [21] Zhanblan-Pike, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Uspekhi Mat. Nauk 50 25–46.
  • [22] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [23] Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44 428–443.
  • [24] Li, S. (2006). The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion. Scand. Actuar. J. 2 73–85.
  • [25] Lin, X. S., Willmot, G. E. and Drekic, S. (2003). The classical risk model with a constant dividend barrier: Analysis of the Gerber–Shiu discounted penalty function. Insurance Math. Econom. 33 551–566.
  • [26] Radner, R. and Shepp, L. (1996). Risk vs. profit potential: A model for corporate strategy. J. Econom. Dynamics Control 20 1373–1393.
  • [27] Rao, M., Song, R. and Vondraček, Z. (2006). Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25 1–27.
  • [28] Renaud, J.-F. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Probab. 44 420–427.
  • [29] Schmidli, H. (2006). Optimisation in non-life insurance. Stoch. Models 22 689–722.
  • [30] Song, R. and Vondraček, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 817–847.
  • [31] Taksar, M. I. (2000). Optimal risk and dividend distribution control models for an insurance company. Math. Methods Oper. Res. 51 1–42.
  • [32] Yuen, K. C., Wang, G. and Li, W. K. (2007). The Gerber–Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier. Insurance Math. Econom. 40 104–112.
  • [33] Zhou, X. (2005). On a classical risk model with a constant dividend barrier. N. Am. Actuar. J. 9 95–108.
  • [34] Zhou, X. (2006). Discussion of “On optimal dividend strategies in the compound Poisson model,” by H. Gerber and E. Shiu. N. Am. Actuar. J. 10 79–84.