# zbMATH — the first resource for mathematics

Optimal dividend and equity issuance problem with proportional and fixed transaction costs. (English) Zbl 1285.91065
Summary: This paper investigates the optimal dividend problem of an insurance company, which controls risk exposure by reinsurance and by issuing new equity to protect from bankruptcy. Transaction costs are incurred by these business activities: reinsurance is non-cheap, dividend is taxed and fixed costs are generated by equity issuance. The goal of the company is to maximize the expected cumulative discounted dividend minus the expected discounted costs of equity issuance. This problem is formulated as a mixed regular-singular-impulse stochastic control problem. By solving the corresponding HJB equation, we obtain the analytical solutions of the optimal return function and the optimal strategy.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control
Full Text:
##### References:
 [1] 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 [2] Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: mathematics and economics, 20, 1, 1-15, (1997) · Zbl 1065.91529 [3] Avanzi, B.; Shen, J.; Wong, B., Optimal dividends and capital injections in the dual model with diffusion, ASTIN bulletin, 41, 2, 611-644, (2011) · Zbl 1242.91089 [4] Avram, F.; Palmowski, Z.; Pistorius, M., On the optimal dividend problem for a spectrally negative Lévy process, The annals of applied probability, 17, 1, 156-180, (2007) · Zbl 1136.60032 [5] Bai, L.; Guo, J.; Zhang, H., Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative finance, 10, 10, 1163-1172, (2010) · Zbl 1208.91060 [6] Borch, K., The mathematical theory of insurance: an annotated selection of papers on insurance published 1960-1972, (1974), Lexington Books [7] 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, 1, 181-202, (2006) · Zbl 1136.91473 [8] Choulli, T.; Taksar, M.; Zhou, X., Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative finance, 1, 6, 573-596, (2001) · Zbl 1405.91251 [9] Choulli, T.; Taksar, M.; Zhou, X., A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM journal on control and optimization, 41, 6, 1946-1982, (2003) · Zbl 1084.91047 [10] Dai, H.; Liu, Z.; Luan, N., Optimal dividend strategies in a dual model with capital injections, Mathematical methods of operations research, 72, 1, 129-143, (2010) · Zbl 1194.91188 [11] Eisenberg, J., On optimal control of capital injections by reinsurance and investments, Blätter der DGVFM, 31, 2, 329-345, (2010) · Zbl 1205.91080 [12] Fleming, W.; Soner, H., Controlled Markov processes and viscosity solutions, vol. 25, (2006), Springer Verlag · Zbl 1105.60005 [13] Grandell, J., Empirical bounds for ruin probabilities, Stochastic processes and their applications, 8, 3, 243-255, (1979) · Zbl 0401.62079 [14] Harrison, J.; Taylor, A., Optimal control of a Brownian storage system, Stochastic processes and their applications, 6, 2, 179-194, (1978) · Zbl 0372.60116 [15] He, L.; Liang, Z., Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: mathematics and economics, 44, 1, 88-94, (2009) · Zbl 1156.91395 [16] Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical finance, 9, 2, 153-182, (1999) · Zbl 0999.91052 [17] Jeanblanc-Picqué, M.; Shiryaev, A., Optimization of the flow of dividends, Russian mathematical surveys, 50, 257, (1995) · Zbl 0878.90014 [18] Karatzas, I.; Shreve, S., Brownian motion and stochastic calculus, vol. 113, (1991), Springer Verlag · Zbl 0734.60060 [19] Kulenko, N.; Schmidli, H., Optimal dividend strategies in a cramér-lundberg model with capital injections, Insurance: mathematics and economics, 43, 2, 270-278, (2008) · Zbl 1189.91075 [20] Løkka, A.; Zervos, M., Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: mathematics and economics, 42, 3, 954-961, (2008) · Zbl 1141.91528 [21] Meng, H.; Siu, T., Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM journal on control and optimization, 49, 1, 254-279, (2011) · Zbl 1229.91164 [22] Protter, P., Stochastic integration and differential equations, vol. 21, (2004), Springer Verlag · Zbl 1041.60005 [23] Scheer, N.; Schmidli, H., Optimal dividend strategies in a cramer – lundberg model with capital injections and administration costs, European actuarial journal, 1, 1, 57-92, (2011) · Zbl 1222.91026 [24] Taksar, M., Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of operations research, 98, 1, 89-99, (2000) · Zbl 1035.91039 [25] Taksar, M.; Zhou, X., Optimal risk and dividend control for a company with a debt liability, Insurance: mathematics and economics, 22, 1, 105-122, (1998) · Zbl 0907.90101 [26] Yao, D.; Yang, H.; Wang, R., Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European journal of operational research, 211, 3, 568-576, (2011) · Zbl 1237.91143
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.