On optimal dividends with exponential and linear penalty payments. (English) Zbl 1394.91235

Summary: We study the optimal dividend problem where the surplus process of an insurance company is modelled by a diffusion process. The insurer is not ruined when the surplus becomes negative, but penalty payments occur, depending on the level of the surplus. The penalty payments shall avoid that losses can rise above any number and can be seen as a preference measure or costs for negative capital. As examples, exponential and linear penalty payments are considered. It turns out that a barrier dividend strategy is optimal.


91B30 Risk theory, insurance (MSC2010)
60J60 Diffusion processes
Full Text: DOI


[1] Albrecher, H.; Bäuerle, N.; Thonhauser, S., Optimal dividend payout in random discrete time, Stat. Risk Model., 28, 251-276, (2011) · Zbl 1233.91139
[2] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation times for the compound Poisson risk model: dividends, Astin Bull., 41, 645-672, (2011) · Zbl 1239.91072
[3] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation times for the compound Poisson risk model: the discounted penalty function, Scand. Actuar. J., 424-452, (2013) · Zbl 1401.91089
[4] Albrecher, H.; Gerber, H. U.; Shiu, S. W., The optimal dividend barrier in the gamma-omega model, Eur. Actuar. J., 1, 43-55, (2011) · Zbl 1219.91062
[5] Albrecher, H.; Lautscham, V., From ruin to bankruptcy for compound Poisson surplus processes, Astin Bull., 43, 213-243, (2013) · Zbl 1283.91084
[6] Avanzi, B., Strategies for dividend distribution: A review, N. Am. Actuar. J., 13, 217-251, (2009)
[7] Azcue, P.; Muler, N., Stochastic optimization in insurance. A dynamic programming approach, (2014), Springer New York · Zbl 1308.91004
[8] de Finetti, B., 1957. Su un’ impostazione alternativa della teoria collettiva del rischio, in: Transactions of th XVth International Congress of Actuaries, Vol. 2, pp. 433-443.
[9] Dickson, D. C.M.; Waters, H. R., Some optimal dividend problems, Astin Bull., 34, 49-74, (2004) · Zbl 1097.91040
[10] Embrechts, P.; Schmidli, H., Ruin estimation for a general insurance risk model, Adv. Appl. Probab., 26, 404-422, (1994) · Zbl 0811.62096
[11] Gerber, H. U., Entscheidungskriterien für den zusammengesetzten Poisson-prozess, Schweiz. Ver. Versicher. Mitt., 69, 185-228, (1969) · Zbl 0193.20501
[12] Gerber, H. U., Der einfluss von zins auf die ruinwahrscheinlichkeit, Schweiz. Ver. Versicher. Mitt., 71, 63-70, (1971) · Zbl 0217.26804
[13] Gerber, H. U.; Shiu, E. S.W., Optimal dividends: analysis with Brownian motion, N. Am. Actuar. J., 8, 1, 1-20, (2004) · Zbl 1085.62122
[14] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance Math. Econom., 27, 215-228, (2000) · Zbl 1007.91025
[15] Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J., 166-180, (1997) · Zbl 1075.91559
[16] Kulenko, N.; Schmidli, H., Optimal dividend strategies in a cramér-lundberg model with capital injections, Insurance Math. Econom., 43, 270-278, (2008) · Zbl 1189.91075
[17] Marciniak, E.; Palmowski, Z., On the optimal dividend problem for insurance risk models with surplus-dependent premiums, J. Optim. Theory Appl., 168, 723-742, (2015) · Zbl 1344.49029
[18] Schmidli, H., Diffusion approximations for a risk process with the possibility of borrowing and investment, Commun. Stat. Stoch. Models, 10, 365-388, (1994) · Zbl 0793.60095
[19] Schmidli, H., On minimising the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12, 890-907, (2002) · Zbl 1021.60061
[20] Schmidli, H., Stochastic control in insurance, (2008), Springer-Verlag Berlin · Zbl 1133.93002
[21] Schmidli, H., On capital injections and dividends with tax in a classical risk model, Insurance Math. Econom, 71, 138-144, (2016) · Zbl 1371.91108
[22] Schmidli, H., On capital injections and dividends with tax in a diffusion approximation, Scand. Actuar. J., (2016), (forthcoming) · Zbl 1371.91108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.