American Institute of Mathematical Sciences

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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

Dingjun Yao 1, , Hailiang Yang 2, and  Rongming Wang 3,

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.
Keywords: Hamilton-Jacobi-Bellman equation., The dual risk model, dividend payment, optimal strategy, proportional transaction costs, equity issuance.
Mathematics Subject Classification: Primary: 93E20, 49L20 Secondary: 91G8.
Citation: Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial & 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.  doi: 10.1007/s007800050075.  Google Scholar

[2]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model,, Insurance: Mathematics and Economics, 41 (2007), 111.  doi: 10.1016/j.insmatheco.2006.10.002.  Google Scholar

[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.  doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[4]

B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio,, Transactions of the XV International Congress of Actuaries, 2 (1957), 433.   Google Scholar

[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.   Google Scholar

[6]

W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (1993).   Google Scholar

[7]

H. U. Gerber, Games of economic survival with discrete- and continuous-income processes,, Operations Research, 20 (1972), 37.  doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.   Google Scholar

[9]

J. Grandell, "Aspects of Risk Theory,", New York, (1991).   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[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.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[12]

M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends,, Russian Mathematical Surveys, 50 (1995), 257.  doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar

[13]

N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections,, Insurance: Mathmatics and Economics, 43 (2008), 270.  doi: 10.1016/j.insmatheco.2008.05.013.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[16]

A. C. Y. Ng, On a dual model with a dividend threshold,, Insurance: Mathematics and Economics, 44 (2009), 315.  doi: 10.1016/j.insmatheco.2008.11.011.  Google Scholar

[17]

H. L. Seal, "Stochastic Theory of a Risk Business,", Wiley, (1969).   Google Scholar

[18]

S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns,, Mathematical Finance, 12 (2002), 155.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2009.5.81.  Google Scholar

[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.  doi: 10.1080/03610920802117080.  Google Scholar

show all references

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.  doi: 10.1007/s007800050075.  Google Scholar

[2]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model,, Insurance: Mathematics and Economics, 41 (2007), 111.  doi: 10.1016/j.insmatheco.2006.10.002.  Google Scholar

[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.  doi: 10.1111/j.1467-9965.2006.00267.x.  Google Scholar

[4]

B. De Finetti, Su un'impostazione alternativa dell teoria colletiva del rischio,, Transactions of the XV International Congress of Actuaries, 2 (1957), 433.   Google Scholar

[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.   Google Scholar

[6]

W. H. Flemming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Springer-Verlag, (1993).   Google Scholar

[7]

H. U. Gerber, Games of economic survival with discrete- and continuous-income processes,, Operations Research, 20 (1972), 37.  doi: 10.1287/opre.20.1.37.  Google Scholar

[8]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion,, North American Actuarial Journal, 8 (2004), 1.   Google Scholar

[9]

J. Grandell, "Aspects of Risk Theory,", New York, (1991).   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[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.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[12]

M. Jeanblanc and A. N. Shiryaev, Optimization of the flow of dividends,, Russian Mathematical Surveys, 50 (1995), 257.  doi: 10.1070/RM1995v050n02ABEH002054.  Google Scholar

[13]

N. Kulenko and H. Schimidli, Optimal dividend strategy in a Cramér-Lundberg model with capital injections,, Insurance: Mathmatics and Economics, 43 (2008), 270.  doi: 10.1016/j.insmatheco.2008.05.013.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[16]

A. C. Y. Ng, On a dual model with a dividend threshold,, Insurance: Mathematics and Economics, 44 (2009), 315.  doi: 10.1016/j.insmatheco.2008.11.011.  Google Scholar

[17]

H. L. Seal, "Stochastic Theory of a Risk Business,", Wiley, (1969).   Google Scholar

[18]

S. P. Sethi and M. Taksar, Optimal financing of a corporation subject to random returns,, Mathematical Finance, 12 (2002), 155.  doi: 10.1111/1467-9965.t01-2-02002.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2009.5.81.  Google Scholar

[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.  doi: 10.1080/03610920802117080.  Google Scholar

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