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Dividend maximization under consideration of the time value of ruin. (English) Zbl 1119.91047
Summary: In the Cramér-Lundberg model and its diffusion approximation, it is a classical problem to find the optimal dividend payment strategy that maximizes the expected value of the discounted dividend payments until ruin. One often raised disadvantage of this approach is the fact that such a strategy does not take the lifetime of the controlled process into account. In this paper we introduce a value function which considers both expected dividends and the time value of ruin. For both the diffusion model and the Cramér-Lundberg model with exponential claim sizes, the problem is solved and in either case the optimal strategy is identified, which for unbounded dividend intensity is a barrier strategy and for bounded dividend intensity is of threshold type.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: mathematics and economics, 20, 1, 1-15, (1997) · Zbl 1065.91529
[2] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the cramér – lundberg model, Mathematical finance, 15, 2, 261-308, (2005) · Zbl 1136.91016
[3] Bäuerle, N., Approximation of optimal reinsurance and dividend payout policies, Mathematical finance, 14, 1, 99-113, (2004) · Zbl 1097.91052
[4] Boguslavskaya, E., 2003. On optimization of dividend flow for a company in the presence of liquidation value. Working Paper
[5] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics operations research, 20, 4, 937-958, (1995) · Zbl 0846.90012
[6] Cont, R., Tankov, P., 2004. Financial Modelling with Jump Processes. In: Chapman & Hall/CRC Financial Mathematics Series, Boca Raton · Zbl 1052.91043
[7] Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1993), Springer New York · Zbl 0773.60070
[8] Gaier, J.; Grandits, P., Ruin probabilities in the presence of regularly varying tails and optimal investment, Insurance: mathematics and economics, 30, 2, 211-217, (2002) · Zbl 1055.91049
[9] Gerber, H.U., Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess, Schweizerische aktuarvereinigung. mitteilungen, 1, 185-227, (1969) · Zbl 0193.20501
[10] Gerber, H.U., Lin, X.S., Yang, H., A note on the dividends-penalty identity and the optimal dividend barrier. In: 10th International Congress on Insurance: Mathematics and Economics, Leuven 2006 (preprint) · Zbl 1162.91374
[11] Gerber, H.U.; Shiu, E.S.W., On optimal dividend strategies in the compound Poisson model, North American actuarial journal, 10, 2, 76-93, (2006)
[12] Grandell, J., A class of approximations of ruin probabilities, Scandinavian actuarial journal, suppl., 37-52, (1977) · Zbl 0384.60057
[13] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance: mathematics and economics, 27, 2, 215-228, (2000) · Zbl 1007.91025
[14] Hipp, C.; Vogt, M., Optimal dynamic XL reinsurance, Astin bulletin, 33, 2, 193-207, (2003) · Zbl 1059.93135
[15] Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical finance, 9, 2, 153-182, (1999) · Zbl 0999.91052
[16] Iglehart, D.L., Diffusion approximations in collective risk theory, Journal of applied probability, 6, 285-292, (1969) · Zbl 0191.51202
[17] Lin, X.S.; Willmot, G.E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the gerber – shiu discounted penalty function, Insurance: mathematics and economics, 33, 3, 551-566, (2003) · Zbl 1103.91369
[18] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., ()
[19] Schmidli, H., Optimal proportional reinsurance policies in a dynamic setting, Scandinavian actuarial journal, 1, 55-68, (2001) · Zbl 0971.91039
[20] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, The annals of applied probability, 12, 3, 890-907, (2002) · Zbl 1021.60061
[21] Schmidli, H., Diffusion approximations, () · Zbl 0814.62066
[22] Schmidli, H., 2006. Optimal Control in Insurance. Springer, Berlin (in press) · Zbl 1107.93036
[23] Shreve, S.E.; Lehoczky, J.P.; Gaver, D.P., Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM journal on control and optimization, 22, 1, 55-75, (1984) · Zbl 0535.93071
[24] Taksar, M.I., Optimal risk and dividend distribution control models for an insurance company, Mathematical methods of operations research, 51, 1, 1-42, (2000) · Zbl 0947.91043
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