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Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. (English) Zbl 1237.91143
Summary: We consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of D. Yao, H. Yang and R. Wang [J. Ind. Manag. Optim. 6, No. 4, 761–777 (2010; Zbl 1218.93112)], we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.

MSC:
91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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[1] Albrecher, H., Thonhauser, S., 2009a. Some remarks on an impulse control problem in insurance. Preprint, University of Lausanne. · Zbl 1187.93138
[2] Albrecher, H.; Thonhauser, S., Optimality results for dividend problems in insurance, revista de la real academia de ciencias exactas, Fisicas y naturales. serie A: matematicas, 103, 2, 295-320, (2009) · Zbl 1187.93138
[3] Asmussen, S.; Hφgaard, B.; Taksar, M., Optimal risk control and dividend distribution policies: example of excess-of-loss reinsurance for an insurance corporation, Finance and stochastics, 4, 299-324, (2000) · Zbl 0958.91026
[4] Avanzi, B.; Gerber, H.U.; Shiu, E.S.W., Optimal dividends in the dual model, Insurance: mathematics and economics, 41, 111-123, (2007) · Zbl 1131.91026
[5] Bai, L.H.; Guo, J.Y., Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian actuarial journal, 1-20, (2008)
[6] Belhaj, M., Optimal dividend payments when cash reserves follow a jump-diffusion process, Mathematical finance, 20, 2, 313-325, (2010) · Zbl 1222.91063
[7] Bensoussan, A.; Lions, J., Impulse control and quasi-variational inequalities, (1984), John Wiley and Sons Ltd
[8] Cadenillas, A.; Zapatero, F., Central bank intervention in the foreign exchange market, Journal of economic theory, 87, 218-242, (1999) · Zbl 0998.91041
[9] 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
[10] Dong, Y.H.; Wang, G.J., Ruin probability for renewal risk model with negative risk sums, Journal of industrial and management optimization, 2, 229-236, (2006) · Zbl 1135.91367
[11] Eisenberg, J.; Schmidli, H., Optimal control of capital injections by reinsurance in a diffusion approximation, Blätter DGVFM, 30, 1-13, (2009) · Zbl 1183.91069
[12] Eisenberg, J., Schmidli, H., 2010. Minimising expected discounted capital injections by reinsurance in a classical risk model. Scandinavian Actuarial Journal, in press. · Zbl 1277.60145
[13] Gerber, H.U.; Shiu, E.S.W., Optimal dividends: analysis with Brownian motion, North American actuarial journal, 8, 1-20, (2004) · Zbl 1085.62122
[14] Gerber, H.U.; Shiu, E.S.W., On optimal dividends strategies in the compound Poisson model, North American acutuarial journal, 10, 2, 76-93, (2006)
[15] Grandell, J., Aspects of risk theory, (1991), Springer-Verlag New York · Zbl 0717.62100
[16] He, L.; Liang, Z.X., Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: mathematics and economics, 42, 88-94, (2009) · Zbl 1156.91395
[17] Hφgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical finance, 9, 153-182, (1999) · Zbl 0999.91052
[18] Hφgaard, B.; Taksar, M., Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy, Quantitative finance, 4, 315-327, (2004) · Zbl 1405.91558
[19] Jeanblanc-Picqué, M.; Shiryaev, A.N., Optimization of the flow of dividends, Russian mathematical surveys, 50, 257-277, (1995) · Zbl 0878.90014
[20] Kulenko, N.; Schimidli, H., An optimal dividend strategy in a cramér – lundberg model with capital injections, Insurance: mathmatics and economics, 43, 270-278, (2008) · Zbl 1189.91075
[21] Loeffen, R.L., An optimal dividends problem with transaction costs for spectrally negative Lévy processes, Insurance: mathematics and economics, 45, 41-48, (2009) · Zbl 1231.91211
[22] Lφkka, A.; Zervos, M., Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: mathematics and economics, 42, 954-961, (2008) · Zbl 1141.91528
[23] Ng, A.C.Y., On a dual model with a dividend threshold, Insurance: mathematics and economics, 44, 315-324, (2009) · Zbl 1163.91441
[24] Ohnishi, M.; Tsujimura, M., An impulse control of a geometric Brownian motion with quadratic costs, European journal of operational reserch, 168, 311-321, (2006) · Zbl 1099.93045
[25] Sethi, S.P.; Taksar, M., Optimal financing of a corporation subject to random returns, Mathematical finance, 12, 155-172, (2002) · Zbl 1048.91068
[26] 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, 55-75, (1984) · Zbl 0535.93071
[27] Yao, D.J.; Yang, H.L.; Wang, R.M., Optimal financing and dividend strategies in a dual model with proportional costs, Journal of industrial and management optimization, 6, 4, 761-777, (2010) · Zbl 1218.93112
[28] Zhu, J.X.; Yang, H.L., Ruin probabilities of a dual Markov-modulated risk model, Communications in statistics-theory and methods, 37, 3298-3307, (2008) · Zbl 1292.91100
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