Optimal dividend and capital injection strategy with a penalty payment at ruin: restricted dividend payments. (English) Zbl 1445.91055

Summary: In this paper, we study the optimal dividend and capital injection problem with the penalty payment at ruin. The dividend strategy is assumed to be restricted to a small class of absolutely continuous strategies with bounded dividend density. By considering the surplus process killed at the time of ruin, we transform the problem to a combined stochastic and impulse control one up to ruin with a free boundary at zero. We illustrate the theoretical verifications for different types of capital injection strategies comparing to the conventional results given in the literature, where the capital injections are made before the time of ruin. Under the assumption of restricted dividend density, the value function is proved as the unique increasing, bounded, Lipschitz continuous and upper semi-continuous at zero viscosity solution to the corresponding quasi-variational Hamilton-Jacobi-Bellman (HJB) equation. The uniqueness of such class of viscosity solutions is shown by considering its boundary condition at infinity. The optimality of a specific band-type strategy is proved for the case when the premium rate is (i) greater than or (ii) less than the ceiling dividend rate respectively. Some numerical examples are presented under the exponential and gamma claim size assumptions.


91G05 Actuarial mathematics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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[1] Albrecher, H.; Thonhauser, S., Optimal dividend strategies for a risk process under force of interest, Insurance Math. Econom., 43, 1, 134-149 (2008) · Zbl 1140.91371
[2] Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance Math. Econom., 20, 1, 1-15 (1997) · Zbl 1065.91529
[3] Avanzi, B.; Gerber, H. U., Optimal dividends in the dual model with diffusion, ASTIN Bull., 38, 2, 653-667 (2008) · Zbl 1274.91463
[4] Avanzi, B.; Gerber, H. U.; Shiu, E. S., Optimal dividends in the dual model, Insurance Math. Econom., 41, 1, 111-123 (2007) · Zbl 1131.91026
[5] Avanzi, B.; Shen, J.; Wong, B., Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41, 2, 611-644 (2011) · Zbl 1242.91089
[6] 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, 4, 1868-1935 (2015) · Zbl 1322.60055
[7] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Finance, 15, 2, 261-308 (2005) · Zbl 1136.91016
[8] Azcue, P.; Muler, N., Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates, Insurance Math. Econom., 51, 1, 26-42 (2012) · Zbl 1284.91201
[9] Azcue, P.; Muler, N., Stochastic Optimization in Insurance: A Dynamic Programming Approach (2014), Springer · Zbl 1308.91004
[10] Bardi, M.; Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (2008), Springer Science & Business Media · Zbl 1134.49022
[11] Chancelier, J.-P.; Messaoud, M.; Sulem, A., A policy iteration algorithm for fixed point problems with nonexpansive operators, Math. Methods Oper. Res., 65, 2, 239-259 (2007) · Zbl 1171.47051
[12] Crandall, M. G.; Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277, 1, 1-42 (1983) · Zbl 0599.35024
[13] De Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2. pp. 433-443.
[14] Dickson, D. C.; Waters, H. R., Some optimal dividends problems, ASTIN Bull., 34, 1, 49-74 (2004) · Zbl 1097.91040
[15] Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Vol. 25 (2006), Springer Science & Business Media · Zbl 1105.60005
[16] Gerber, H. U., Entscheidungskriterien für den zusammengesetzten Poisson-Prozess (1969), (Ph.D. thesis) · Zbl 0193.20501
[17] Gerber, H. U.; Shiu, E. S., On optimal dividend strategies in the compound Poisson model, N. Am. Actuar. J., 10, 2, 76-93 (2006)
[18] Gerber, H. U.; Shiu, E. S.; Smith, N., Maximizing dividends without bankruptcy, ASTIN Bull., 36, 1, 5-23 (2006) · Zbl 1162.91375
[19] Hernandez, C.; Junca, M.; Moreno-Franco, H., A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes, Insurance Math. Econom., 79, 1, 57-68 (2018) · Zbl 1401.91147
[20] Jin, Z.; Yang, H.; Yin, G., A numerical approach to optimal dividend policies with capital injections and transaction costs, Acta Math. Appl. Sin. Engl. Ser., 33, 1, 221-238 (2017) · Zbl 1360.91153
[21] Kulenko, N.; Schmidli, H., Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance Math. Econom., 43, 2, 270-278 (2008) · Zbl 1189.91075
[22] Kyprianou, A. E.; Loeffen, R.; Pérez, J.-L., Optimal control with absolutely continuous strategies for spectrally negative Lévy processes, J. Appl. Probab., 49, 1, 150-166 (2012) · Zbl 1253.93001
[23] Liang, Z.; Young, V. R., Dividends and reinsurance under a penalty for ruin, Insurance Math. Econom., 50, 3, 437-445 (2012) · Zbl 1236.91086
[24] Loeffen, R., An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Probab., 46, 1, 85-98 (2009) · Zbl 1166.60051
[25] Loeffen, R. L.; Renaud, J.-F., De Finetti’s optimal dividends problem with an affine penalty function at ruin, Insurance Math. Econom., 46, 1, 98-108 (2010) · Zbl 1231.91212
[26] Nie, C.; Dickson, D. C.; Li, S., The finite time ruin probability in a risk model with capital injections, Scand. Actuar. J., 2015, 4, 301-318 (2015) · Zbl 1398.91350
[27] Øksendal, B. K.; Sulem, A., Applied Stochastic Control of Jump Diffusions, Vol. 498 (2005), Springer
[28] Peng, X.; Chen, M.; Guo, J., Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance Math. Econom., 51, 3, 576-585 (2012) · Zbl 1285.91065
[29] Rogers, L. C., Optimal Investment (2013), Springer · Zbl 1264.91119
[30] Scheer, N.; Schmidli, H., Optimal dividend strategies in a Cramer-Lundberg model with capital injections and administration costs, Eur. Actuar. J., 1, 1, 57-92 (2011) · Zbl 1222.91026
[31] Schmidli, H., Stochastic Control in Insurance (2008), Springer Science & Business Media · Zbl 1133.93002
[32] Thonhauser, S.; Albrecher, H., Dividend maximization under consideration of the time value of ruin, Insurance Math. Econom., 41, 1, 163-184 (2007) · Zbl 1119.91047
[33] Thonhauser, S.; Albrecher, H., Optimal dividend strategies for a compound Poisson process under transaction costs and power utility, Stoch. Models, 27, 1, 120-140 (2011) · Zbl 1262.91096
[34] Vierkötter, M.; Schmidli, H., On optimal dividends with exponential and linear penalty payments, Insurance Math. Econom., 72, 265-270 (2017) · Zbl 1394.91235
[35] Xu, R.; Woo, J.-K.; Han, X.; Yang, H., A plan of capital injections based on the claims frequency, Ann. Actuar. Sci., 12, 2, 296-325 (2018)
[36] Yao, D.; Yang, H.; Wang, R., Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European J. Oper. Res., 211, 3, 568-576 (2011) · Zbl 1237.91143
[37] Zhang, Z.; Cheung, E. C.; Yang, H., On the compound Poisson risk model with periodic capital injections, ASTIN Bull., 48, 1, 435-477 (2018) · Zbl 1390.91220
[38] Zhao, Y.; Chen, P.; Yang, H., Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes, Insurance Math. Econom., 74, 135-146 (2017) · Zbl 1394.91243
[39] Zhao, Y.; Wang, R.; Yin, C., Optimal dividends and capital injections for a spectrally positive Lévy process, J. Ind. Manag. Optim., 13, 1-21 (2017) · Zbl 1362.93171
[40] Zhu, J.; Yang, H., Optimal financing and dividend distribution in a general diffusion model with regime switching, Adv. Appl. Probab., 48, 2, 406-422 (2016) · Zbl 1343.49032
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