The Annals of Applied Probability

On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function

F. Avram, Z. Palmowski, and M. R. Pistorius

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This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative Lévy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber–Shiu function. A number of concrete examples are analyzed.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 1868-1935.

Received: December 2012
Revised: May 2014
First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

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

Stochastic control singular control impulse control state-constraint problem stochastic solution integro-differential HJB equation Lévy process De Finetti model barrier/band strategy Gerber–Shiu function


Avram, F.; Palmowski, Z.; Pistorius, M. R. On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function. Ann. Appl. Probab. 25 (2015), no. 4, 1868--1935. doi:10.1214/14-AAP1038.

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  • [1] Albrecher, H. and Thonhauser, S. (2008). Optimal dividend strategies for a risk process under force of interest. Insurance Math. Econom. 43 134–149.
  • [2] Alvarez, L. H. R. and Virtanen, J. (2006). A class of solvable stochastic dividend optimization problems: On the general impact of flexibility on valuation. Econom. Theory 28 373–398.
  • [3] Alvarez, O. and Tourin, A. (1996). Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 293–317.
  • [4] 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.
  • [5] Avram, F., Palmowski, Z. and Pistorius, M. (2010). On optimal dividend distribution for a Cramér–Lundberg process with exponential jumps in the presence of a linear Gerber–Shiu penalty function. In The Pyrenees International Workshop and Summer School on Statistics, Probability and Operations Research—SPO 2009. Monogr. Mat. García Galdeano 36 69–77. Prensas Univ. Zaragoza, Zaragoza.
  • [6] 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.
  • [7] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 261–308.
  • [8] Azcue, P. and Muler, N. (2012). Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates. Insurance Math. Econom. 51 26–42.
  • [9] Bardi, M. and Capuzzo-Dolcetta, I. (1997). Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston, MA.
  • [10] Bayraktar, E. and Sîrbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case. Proc. Amer. Math. Soc. 140 3645–3654.
  • [11] Benth, F. E., Karlsen, K. H. and Reikvam, K. (2001). Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach. Finance Stoch. 5 275–303.
  • [12] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [13] Biffis, E. and Kyprianou, A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46 85–91.
  • [14] Cai, J., Feng, R. and Willmot, G. E. (2009). On the expectation of total discounted operating costs up to default and its applications. Adv. in Appl. Probab. 41 495–522.
  • [15] De Finetti, B. (1957). Su un’impostazione alternativa dell teoria colletiva del rischio. Transactions of the XV International Congress of Actuaries 2 433–443.
  • [16] Dickson, D. C. M. and Waters, H. R. (2004). Some optimal dividends problems. Astin Bull. 34 49–74.
  • [17] Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (New York) 25. Springer, New York.
  • [18] Gerber, H. U. (1969). Entscheidungskriterien für den Zusammengesetzten Poisson Prozess. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker 69 185–228.
  • [19] Gerber, H. U. (1972). Games of economic survival with discrete- and continuous-income processes. Oper. Res. 20 37–45.
  • [20] Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. S.S. Heubner Foundation Monograph Series 8. S.S. Huebner Foundation for Insurance Education, Philadelphia, PA.
  • [21] 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.
  • [22] Hallin, M. (1979). Band strategies: The random walk of reserves. Blätter der DGVFM 14 321–236.
  • [23] Kulenko, N. and Schmidli, H. (2008). Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insurance Math. Econom. 43 270–278.
  • [24] 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 in Math. 2061 97–186. Springer, Heidelberg.
  • [25] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [26] Kyprianou, A. E., Rivero, V. and Song, R. (2010). Convexity and smoothness of scale functions and de Finetti’s control problem. J. Theoret. Probab. 23 547–564.
  • [27] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18 1669–1680.
  • [28] Loeffen, R. L. (2009). An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insurance Math. Econom. 45 41–48.
  • [29] Loeffen, R. L. and Renaud, J.-F. (2010). De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46 98–108.
  • [30] Neveu, J. (1975). Discrete-parameter Martingales, Revised ed. North-Holland, Amsterdam.
  • [31] Nguyen-Ngoc, L. and Yor, M. (2005). Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 42–69. Springer, Berlin.
  • [32] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhauser, Basel.
  • [33] Pham, H. (1998). Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Systems Estim. Control 8 1–27 pp. (electronic).
  • [34] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Probab. 17 183–220.
  • [35] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [36] Sayah, A. (1991). Equations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels, I; II. Comm. Partial Differential Equations 16 1057–1074, 1075–1093.
  • [37] Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.
  • [38] Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optim. 22 55–75.
  • [39] Soner, H. M. (1986). Optimal control with state-space constraint. I, II. SIAM J. Control Optim. 24 552–561, 1110–1122.
  • [40] Stroock, D. and Varadhan, S. R. S. (1972). On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math. 25 651–713.
  • [41] Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41 163–184.
  • [42] Thonhauser, S. and Albrecher, H. (2011). Optimal dividend strategies for a compound Poisson process under transaction costs and power utility. Stoch. Models 27 120–140.
  • [43] Zajic, T. (2000). Optimal dividend payout under compound Poisson income. J. Optim. Theory Appl. 104 195–213.