# American Institute of Mathematical Sciences

October  2010, 6(4): 761-777. doi: 10.3934/jimo.2010.6.761

## Optimal financing and dividend strategies in a dual model with proportional costs

 1 School of Finance, Nanjing University of Finance and Economics, Nanjing, 210046, China 2 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong 3 School of Finance and Statistics, East China Normal University, Shanghai, 200241

Received  August 2009 Revised  April 2010 Published  September 2010

We consider the optimal control problem with dividend payments and issuance of equity 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. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.
Citation: Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial and Management Optimization, 2010, 6 (4) : 761-777. doi: 10.3934/jimo.2010.6.761
##### References:
 [1] S. Asmussen, B. Høgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [2] B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123. doi: 10.1016/j.insmatheco.2006.10.002. [3] A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse Stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202. doi: 10.1111/j.1467-9965.2006.00267.x. [4] B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2 (1957), 433-443. [5] Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236. [6] W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Springer-Verlag, NewYork, 1993. [7] H. U. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Research, 20 (1972), 37-45. doi: 10.1287/opre.20.1.37. [8] H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. [9] J. Grandell, "Aspects of Risk Theory," New York, Springer-Verlag, 1991. [10] L. He and Z. X. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance strategy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003. [11] B. Høgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007. [12] M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277. doi: 10.1070/RM1995v050n02ABEH002054. [13] N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] G. Lu, Q. Hu, Y. Zhou and W. Yue, Optimal execution strategy with an endogenously determined sales period, Journal of Industrial and Management Optimization, 1 (2005), 280-304. [15] A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [16] A. C. Y. Ng, On a dual model with a dividend threshold, Insurance: Mathematics and Economics, 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011. [17] H. L. Seal, "Stochastic Theory of a Risk Business," Wiley, New York, 1969. [18] S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns, Mathematical Finance, 12 (2002), 155-172. doi: 10.1111/1467-9965.t01-2-02002. [19] L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815. [20] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolio under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. doi: 10.3934/jimo.2009.5.81. [21] J. X. Zhu and H. L. Yang, Ruin probabilities of a dual Markov-modulated risk model, Communications in Statistics-Theory and Methods, 37 (2008), 3298-3307. doi: 10.1080/03610920802117080.

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##### References:
 [1] S. Asmussen, B. Høgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [2] B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123. doi: 10.1016/j.insmatheco.2006.10.002. [3] A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse Stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202. doi: 10.1111/j.1467-9965.2006.00267.x. [4] B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2 (1957), 433-443. [5] Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236. [6] W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Springer-Verlag, NewYork, 1993. [7] H. U. Gerber, Games of economic survival with discrete- and continuous-income processes, Operations Research, 20 (1972), 37-45. doi: 10.1287/opre.20.1.37. [8] H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. [9] J. Grandell, "Aspects of Risk Theory," New York, Springer-Verlag, 1991. [10] L. He and Z. X. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance strategy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003. [11] B. Høgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007. [12] M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends, Russian Mathematical Surveys, 50 (1995), 257-277. doi: 10.1070/RM1995v050n02ABEH002054. [13] N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] G. Lu, Q. Hu, Y. Zhou and W. Yue, Optimal execution strategy with an endogenously determined sales period, Journal of Industrial and Management Optimization, 1 (2005), 280-304. [15] A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [16] A. C. Y. Ng, On a dual model with a dividend threshold, Insurance: Mathematics and Economics, 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011. [17] H. L. Seal, "Stochastic Theory of a Risk Business," Wiley, New York, 1969. [18] S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns, Mathematical Finance, 12 (2002), 155-172. doi: 10.1111/1467-9965.t01-2-02002. [19] L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815. [20] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolio under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. doi: 10.3934/jimo.2009.5.81. [21] J. X. Zhu and H. L. Yang, Ruin probabilities of a dual Markov-modulated risk model, Communications in Statistics-Theory and Methods, 37 (2008), 3298-3307. doi: 10.1080/03610920802117080.
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