This paper concerns an optimal dividend-penalty problem for the risk models with surplus-dependent premiums. The objective is to maximize the difference of the expected cumulative discounted dividend payments received until the moment of ruin and a discounted penalty payment taken at the moment of ruin. Since the value function may be not smooth enough to be the classical solution of the HJB equation, the viscosity solution is involved. The optimal value function can be characterized as the smallest viscosity supersolution of the HJB equation and the optimal dividend-penalty strategy has a band structure. Finally, some numerical examples with gamma distribution for the claims are analyzed.
Jingwei LI
,
Guoxin LIU
,
Jinyan ZHAO
. OPTIMAL DIVIDEND-PENALTY STRATEGIES FOR INSURANCE RISK MODELS WITH SURPLUS-DEPENDENT PREMIUMS[J]. Acta mathematica scientia, Series B, 2020
, 40(1)
: 170
-198
.
DOI: 10.1007/s10473-020-0112-1
[1] Albrecher H, Thonhauser S. Optimal dividend strategies for a risk process under force of interest. Insurance Mathematics and Economics, 2008, 43(1):134-149
[2] 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. Annals of Applied Probability, 2015, 25(4):1868-1935
[3] Azcue P, Muler N. Optimal reinsurance and dividend distribution policies in the Crámer-Lundberg model. Mathematical Finance, 2005, 15(2):261-308
[4] Azcue P, Muler N. Stochastic Optimization in Insurance:a Dynamic Programming Approach. London:Springer, 2014
[5] Benth F E, Karlsen K H, Reikvam K. Optimal portfolio selection with consumption and nonlinear integrodifferential equations with gradient constraint:a viscosity solution approach. Finance and Stochastics, 2001, 5(3):275-303
[6] Cai J, Feng R, Willmot G E. On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability, 2009, 41(2):495-522
[7] Dickson, David C M, Waters H R. Some optimal dividends problems. Astin Bulletin, 2004, 34(1):49-74
[8] Feng R, Volkmer H W, Zhang S, Zhu C. Optimal dividend policies for piecewise-deterministic compound Poisson risk models. Scandinavian Actuarial Journal, 2015, 51(1):423-454
[9] Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics. New York:Springer-Verlag, 1993
[10] Gerber H U. Entscheidungskriterien Füer den Zusammengesetzten Poisson-Proze. Schweiz Verein Versicherungsmath Mitt, 1969
[11] Gerber H U, Lin X S, Yang H. A note on the dividends-penalty identity and the optimal dividend barrier. Astin Bulletin, 2006, 36(2):489-503
[12] Marciniak E, Palmowski Z. On the optimal dividend problem for insurance risk models with surplusdependent premiums. Journal of Optimization Theory and Applications, 2016, 168(2):723-742
[13] Schmidli H. Stochastic Control in Insurance. London:Springer, 2008
[14] Thonhauser S, Albrecher H. Dividend maximization under consideration of the time value of ruin. Insurance Mathematics and Economics, 2007, 41(1):163-184
[15] Zajic T. Optimal dividend payout under compound poisson income. Journal of Optimization Theory and Applications, 2000, 104(1):195-213
[16] Zhu J. Optimal dividend control for a generalized risk model with investment incomes and debit interest. Scandinavian Actuarial Journal, 2013, 2(2):140-162
[17] Liu J, Xu J C, Hu Y J. On the expected discounted penalty function in a Markov-dependent risk model with constant dividend barrier. Acta Mathematics Scientia, 2010, 30B(5):1481-1491
