×

A note on optimal expected utility of dividend payments with proportional reinsurance. (English) Zbl 1416.91201

Summary: In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company that controls risk exposure by purchasing proportional reinsurance. We assume the preference of the insurer is of CRRA form. By solving the corresponding Hamilton-Jacobi-Bellman equation, we identify the value function and the corresponding optimal strategy. We also analyze the asymptotic behavior of the value function for large initial reserves. Finally, we provide some numerical examples to illustrate the results and analyze the sensitivity of the parameters.

MSC:

91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albrecher, H.; Thonhauser, S., Optimality results for dividend problems in insurance. RACSAM - revista de la real academia de ciencias exactas, Fisicas y Naturales. Serie A. Matematicas, 103, 2, 295-320, (2009)
[2] Asmussen, S.; Højgaard, B.; Taksar, M., Optimal risk control and dividend distribution policies. example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4, 3, 299-324, (2000) · Zbl 0958.91026
[3] Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20, 1, 1-15, (1997) · Zbl 1065.91529
[4] Avram, F.; Palmowski, Z.; Pistorius, M. R., On the optimal dividend problem for a spectrally negative Lévy process, Annals of Applied Probability, 17, 156-180, (2007) · Zbl 1136.60032
[5] 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, 25, 4, 1868-1935, (2015) · Zbl 1322.60055
[6] 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
[7] Azcue, P.; Muler, N., Optimal dividend payment and regime switching in a compound Poisson risk model, SIAM Journal on Control and Optimization, 53, 5, 3270-3298, (2015) · Zbl 1372.91052
[8] Bai, L.; Guo, J., Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian Actuarial Journal, 2010, 1, 36-55, (2010) · Zbl 1224.91043
[9] Buhlmann, H., Mathematical methods in risk theory, (1970), Springer-Verlag, Berlin · Zbl 0209.23302
[10] Cadenillas, A.; Choulli, T.; Taksar, M.; Zhang, L., Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16, 181-202, (2006) · Zbl 1136.91473
[11] Chen, M.; Peng, X.; Guo, J., Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52, 3, 448-456, (2013) · Zbl 1284.91213
[12] Choulli, T.; Taksar, M.; Zhou, X. Y., A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41, 6, 1946-1979, (2003) · Zbl 1084.91047
[13] Czarna, I.; Palmowski, Z., Dividend problem with Parisian delay for a spectrally negative Lévy risk process, Journal of Optimization Theory and Applications, 161, 239-256, (2014) · Zbl 1296.91150
[14] David Promislow, S.; Young, V. R., Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9, 3, 110-128, (2005) · Zbl 1141.91543
[15] Fleming, W. H.; Soner, H. M., Controlled Markov processes and viscosity solutions, 25, (2006), Springer, New York · Zbl 1105.60005
[16] Gerber, H. U.; Shiu, E. S. W., Optimal dividends: analysis with Brownian motion, North American Actuarial Journal, 8, 1-20, (2004) · Zbl 1085.62122
[17] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance: Mathematics and Economics, 27, 215-228, (2000) · Zbl 1007.91025
[18] Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical Finance, 9, 2, 153-182, (1999) · Zbl 0999.91052
[19] Højgaard, B.; Taksar, M., Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4, 3, 315-327, (2004)
[20] Hubalek, F.; Schachermayer, W., Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE, Insurance: Mathematics and Economics, 34, 2, 193-225, (2004) · Zbl 1136.91481
[21] Jeanblanc, M.; Shiryaev, A. N., Optimization of the flow of dividends, Russian Mathematical Surveys, 50, 257-277, (1995) · Zbl 0878.90014
[22] Karatzas, I.; Lehoczky, J. P.; Sethi, S. P.; Shreve, S. E., Explicit solution of a general consumption/investment problem, Mathematics of Operations Research, 11, 2, 261-294, (1986) · Zbl 0622.90018
[23] Kyprianou, A.; Palmowski, Z., Distributional study of de finetti’s dividend problem for a general Lévy insurance risk process, Journal of Applied Probability, 44, 428-443, (2007) · Zbl 1137.60047
[24] Liang, Z.; Young, V. R., Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50, 3, 437-445, (2012) · Zbl 1236.91086
[25] Loeffen, R., On optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes, Annals of Applied Probability, 18, 1669-1680, (2008) · Zbl 1152.60344
[26] Loeffen, R., An optimal dividends problem with transaction costs for spectrally negative Lévy processes, Insurance: Mathematics and Economics, 45, 1, 41-48, (2009) · Zbl 1231.91211
[27] Loeffen, R.; Renaud, J.-F., De finetti’s optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46, 98-108, (2009) · Zbl 1231.91212
[28] Merton, R. C., Lifetime portfolio selection under uncertainty: the continuous-time case, The review of Economics and Statistics, 51, 3, 247-257, (1969)
[29] Paulsen, J., Optimal dividend payouts for diffusions with solvency constraints, Finance and Stochastics, 7, 4, 457-473, (2003) · Zbl 1038.60081
[30] Paulsen, J., Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs, SIAM Journal on Control and Optimization, 47, 5, 2201-2226, (2008) · Zbl 1171.49027
[31] Paulsen, J.; Gjessing, H. K., Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: Mathematics and Economics, 20, 215-223, (1997) · Zbl 0894.90048
[32] Schmidli, H., Stochastic control in insurance, (2008), Springer-Verlag, London · Zbl 1133.93002
[33] Zhou, X., On a classical risk model with a constant dividend barrier, North American Actuarial Journal, 9, 1-14, (2005)
[34] Zhou, M.; Yuen, K. C., Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle, Economic Modelling, 29, 2, 198-207, (2012)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.