The Annals of Applied Probability

Optimal dividend and investment problems under Sparre Andersen model

Lihua Bai, Jin Ma, and Xiaojing Xing

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In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated HJB equation on a cylindrical domain on which the problem is well defined.

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Ann. Appl. Probab., Volume 27, Number 6 (2017), 3588-3632.

Received: June 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

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Primary: 91B30: Risk theory, insurance 93E20: Optimal stochastic control 60K05: Renewal theory 35D40: Viscosity solutions

Optimal dividend problem Sparre Andersen model backward Markovization dynamic programming Hamilton–Jacobi–Bellman equation constrained viscosity solution


Bai, Lihua; Ma, Jin; Xing, Xiaojing. Optimal dividend and investment problems under Sparre Andersen model. Ann. Appl. Probab. 27 (2017), no. 6, 3588--3632. doi:10.1214/17-AAP1288.

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  • [1] Albrecher, H., Claramunt, M. M. and Mármol, M. (2005). On the distribution of dividend payments in a Sparre Andersen model with generalized $\mathrm{Erlang}(n)$ interclaim times. Insurance Math. Econom. 37 324–334.
  • [2] Albrecher, H. and Hartinger, J. (2006). On the non-optimality of horizontal dividend barrier strategies in the Sparre Andersen model. Hermis J. Comp. Math. Appl. 7 109–122.
  • [3] Albrecher, H. and Thonhauser, S. (2008). Optimal dividend strategies for a risk process under force of interest. Insurance Math. Econom. 43 134–149.
  • [4] Albrecher, H. and Thonhauser, S. (2009). Optimality results for dividend problems in insurance. Rev. R. Acad. Cienc. Exactas FíS. Nat. Ser. A Math. RACSAM 103 295–320.
  • [5] Alvarez, O. and Tourin, A. (1996). Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 293–317.
  • [6] Asmussen, S., Hojgaard, 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.
  • [7] Awatif, S. (1991). Équations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. Comm. Partial Differential Equations 16 1057–1074.
  • [8] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 261–308.
  • [9] Azcue, P. and Muler, N. (2010). Optimal investment policy and dividend payment strategy in an insurance company. Ann. Appl. Probab. 20 1253–1302.
  • [10] Bai, L., Guo, J. and Zhang, H. (2010). Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes. Quant. Finance 10 1163–1172.
  • [11] Bai, L., Hunting, M. and Paulsen, J. (2012). Optimal dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints. Finance Stoch. 16 477–511.
  • [12] Bai, L. and Paulsen, J. (2010). Optimal dividend policies with transaction costs for a class of diffusion processes. SIAM J. Control Optim. 48 4987–5008.
  • [13] 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.
  • [14] Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M. and Martellini, L. (2008). Optimal investment decisions when time-horizon is uncertain. J. Math. Econom. 44 1100–1113.
  • [15] Browne, S. (1995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 937–958.
  • [16] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
  • [17] De Finetti, B. (1957). Su un’ impostazione alternativa dell teoria collettiva del risichio. Transactions of the XVth Congress of Actuaries (II) 433–443.
  • [18] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
  • [19] Gerber, H. U. (1969). Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1 185–227.
  • [20] Gerber, H. U. and Shiu, E. S. W. (2006). On optimal dividend strategies in the compound Poisson model. N. Am. Actuar. J. 10 76–93.
  • [21] Gordon, M. J. (1959). Dividends, earnings and stock prices. Rev. Econ. Stat. 41 99–105.
  • [22] Hipp, C. and Plum, M. (2000). Optimal investment for insurers. Insurance Math. Econom. 27 215–228.
  • [23] Hipp, C. and Vogt, M. (2003). Optimal dynamic XL reinsurance. Astin Bull. 33 193–207.
  • [24] Hojgaard, B. and Taksar, M. (1998). Optimal proportional reinsurance policies for diffusion models. Scand. Actuar. J. 2 166–180.
  • [25] Hojgaard, B. and Taksar, M. (1999). Controlling risk exposure and dividends payout schemes: Insurance company example. Math. Finance 9 153–182.
  • [26] Ishii, H. and Lions, P.-L. (1990). Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 26–78.
  • [27] Jeanblanc-Picque, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Uspekhi Mat. Nauk 50 25–46.
  • [28] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [29] Li, S. and Garrido, J. (2004). On a class of renewal risk models with a constant dividend barrier. Insurance Math. Econom. 35 691–701.
  • [30] Liu, Y. and Ma, J. (2009). Optimal reinsurance/investment problems for general insurance models. Ann. Appl. Probab. 19 1495–1528.
  • [31] 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.
  • [32] Ma, J. and Sun, X. (2003). Ruin probabilities for insurance models involving investments. Scand. Actuar. J. 3 217–237.
  • [33] Ma, J. and Yu, Y. (2006). Principle of equivalent utility and universal variable life insurance pricing. Scand. Actuar. J. 2006 311–337.
  • [34] Miller, M. H. and Modigliani, F. (1961). Dividend policy, growth, and the valuation of shares. The Journal of Business 34 411–433.
  • [35] Mnif, M. and Sulem, A. (2005). Optimal risk control and dividend policies under excess of loss reinsurance. Stochastics 77 455–476.
  • [36] Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [37] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. Wiley, Chichester.
  • [38] Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuar. J. 2001 55–68.
  • [39] Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.
  • [40] Soner, H. M. (1986). Optimal control with state-space constraint. I. SIAM J. Control Optim. 24 552–561.
  • [41] Sparre Andersen, E. (1957). On the collective theory of risk in the case of contagion between the claims. In Transactions of the XVth Int. Congress of Actuaries, New York, (II) 219–229.
  • [42] Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Applications of Mathematics (New York) 43. Springer, New York.