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Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. (English) Zbl 1156.91395
Summary: We consider the optimal financing and dividend control problem of the insurance company with fixed and proportional transaction costs. The management of the company controls the reinsurance rate, dividends payout as well as the equity issuance process to maximize the expected present value of the dividends payout minus the equity issuance until the time of bankruptcy. This is the first time that the financing process in an insurance model with two kinds of transaction costs, which come from real financial market has been considered. We solve the mixed classical-impulse control problem by constructing two categories of suboptimal models, one is the classical model without equity issuance, the other never goes bankrupt by equity issuance.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91G10 Portfolio theory 91B16 Utility theory 60H05 Stochastic integrals 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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##### References:
 [1] Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: mathematics and economics, 20, 1-15, (1997) · Zbl 1065.91529 [2] Borch, K., The theory of risk, Journal of the royal statistical society. series B, 29, 432-452, (1967) · Zbl 0153.49301 [3] Borch, K., The capital structure of a firm, Swedish journal of economics, 71, 1-13, (1969) [4] Cadenillas, A.; Choulli, T.; Taksar, M.; Zhang, Lei., 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 [5] Cont, R.; Tankov, P., () [6] Gerber, H.U., Games of economic survival with discrete and continuous income processes, Operations research, 20, 37-45, (1972) · Zbl 0236.90079 [7] Harrison, J.M.; Taksar, M.J., Instant control of Brownian motion, Mathematics of operations research, 8, 439-453, (1983) · Zbl 0523.93068 [8] He, Lin; Liang, Zongxia, Optimal financing and dividend control of the insurance company with proportional reinsurance policy, Insurance: mathematics and economics, 42, 976-983, (2007) · Zbl 1141.91445 [9] Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies for diffusion models, Scandinavian actuarial journal, 2, 166-180, (1998) · Zbl 1075.91559 [10] Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies with transaction costs, Insurance: mathematics and economics, 22, 41-51, (1998) · Zbl 1093.91518 [11] Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical finance, 9, 2, 153-182, (1999) · Zbl 0999.91052 [12] Højgaard, B.; Taksar, M., Optimal risk control for a large corporation in the presence of returns on investments, Finance and stochastics, 5, 527-547, (2001) · Zbl 1049.93090 [13] Lions, P.-L.; Sznitman, A.S., Stochastic differential equations with reflecting boundary conditions, Communications on pure and applied mathematics, 37, 511-537, (1984) · Zbl 0598.60060 [14] Martin-Löf, A., Premium control in an insurance system, an approach using linear control theory, Scandinavian actuarial journal, 1-27, (1983) · Zbl 0509.62097 [15] Sethi, S.P.; Taksar, M., Optimal financing of a corporation subject to random returns, Mathematical finance, 12, 2, 155-172, (2002) · Zbl 1048.91068 [16] Taksar, M., Optimal risk/dividend distribution control models: applications to insurance, Mathematical methods of operations research, 1, 1-42, (2000) · Zbl 0947.91043 [17] Wendell, H.; Fleming, H.Mete Soner., Controlled Markov processes and viscosity solutions, ISBN: 0387979271, (1993), Springer-Verlag New York · Zbl 0773.60070
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