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October  2015, 11(4): 1247-1262. doi: 10.3934/jimo.2015.11.1247

Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs

Chuancun Yin 1, and  Kam Chuen Yuen 2,

1. 

School of Statistics, Qufu Normal University, Shandong 273165, China
2. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  April 2014 Revised  February 2015 Published  March 2015

In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.
Keywords: dual model, stochastic control., optimal dividend strategy, HJB equation, jump-diffusion, Barrier strategy.
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 60J7.
Citation: Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, Dover Publications, (1992).   Google Scholar

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models,, Stochastic Processes and their Applications, 109 (2004), 79.  doi: 10.1016/j.spa.2003.07.005.  Google Scholar

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B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency,, Insurance: Mathematics and Economics, 52 (2013), 98.  doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

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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

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B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion,, ASTIN Bulletin, 41 (2011), 611.  doi: 10.2139/ssrn.1709174.  Google Scholar

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B. Avanzi, V. Tu and B. Wong, On optimal periodic dividend strategies in the dual model with diffusion,, Insurance: Mathematics and Economics, 55 (2014), 210.  doi: 10.1016/j.insmatheco.2014.01.005.  Google Scholar

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P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model,, Mathematical Finance, 15 (2005), 261.  doi: 10.1111/j.0960-1627.2005.00220.x.  Google Scholar

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E. Bayraktar and M. Egami, Optimizing venture capital investments in a jump diffusion model,, Mathematical Methods of Operations Research, 67 (2008), 21.  doi: 10.1007/s00186-007-0181-x.  Google Scholar

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[12]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs,, Insurance: Mathematics and Economics, 54 (2014), 133.  doi: 10.1016/j.insmatheco.2013.11.007.  Google Scholar

[13]

E. C. K. Cheung and S. Drekic, Dividend moments in the dual model: Exact and approximate approaches,, ASTIN Bulletin, 38 (2008), 399.  doi: 10.2143/AST.38.2.2033347.  Google Scholar

[14]

H. Dai, Z. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections,, Mathematical Methods of Operations Research, 72 (2010), 129.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[15]

H. Dai, Z. Liu and N. Luan, Optimal financing and dividend control in the dual model,, Mathematical and Computer Modelling, 53 (2011), 1921.  doi: 10.1016/j.mcm.2011.01.019.  Google Scholar

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W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Applications of Mathematics, (1993).   Google Scholar

[18]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs,, Insurance: Mathematics and Economics, 44 (2009), 88.  doi: 10.1016/j.insmatheco.2008.10.001.  Google Scholar

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S. Jaschke, A note on the inhomogeneous linear stochastic differential equation,, Insurance: Mathematics and Economics, 32 (2003), 461.  doi: 10.1016/S0167-6687(03)00134-3.  Google Scholar

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

[21]

K. Miyasawa, An economic survival game,, Journal of the Operations Research Society of Japan, 4 (1962), 95.   Google Scholar

[22]

H. Schmidli, Stochastic Control in Insurance,, Springer, (2008).   Google Scholar

[23]

D. J. Yao, H. L. Yang and R. M. Wang, Optimal financing and dividend strategies in a dual model with proportional costs,, Journal of Industrial and Management Optimization, 6 (2010), 761.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[24]

D. J. Yao, H. L. Yang and R. W. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs,, European Journal of Operational Research, 211 (2011), 568.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

[25]

D. J. Yao, R. W. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model,, Journal of Industrial and Management Optimization, 10 (2014), 1235.  doi: 10.3934/jimo.2014.10.1235.  Google Scholar

[26]

C. C. Yin and Y. Z. Wen, Optimal dividends problem with a terminal value for spectrally positive Lévy processes,, Insurance: Mathematics and Economics, 53 (2013), 769.  doi: 10.1016/j.insmatheco.2013.09.019.  Google Scholar

[27]

C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments,, Insurance: Mathematics and Economics, 52 (2013), 469.  doi: 10.1016/j.insmatheco.2013.02.014.  Google Scholar

[28]

C. C. Yin, Y. Z. Wen and Y. X. Zhao, On the optimal dividend problem for a spectrally positive Lévy process,, ASTIN Bulletin, 44 (2014), 635.  doi: 10.1017/asb.2014.12.  Google Scholar

[29]

Z. M. Zhang, On a risk model with randomized dividend-decision times,, Journal of Industrial and Management Optimization, 10 (2014), 1041.  doi: 10.3934/jimo.2014.10.1041.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, Dover Publications, (1992).   Google Scholar

[2]

S. Asmussen, F. Avram and M. R. Pistorius, Russian and American put options under exponential phase-type Lévy models,, Stochastic Processes and their Applications, 109 (2004), 79.  doi: 10.1016/j.spa.2003.07.005.  Google Scholar

[3]

B. Avanzi, Strategies for dividend distribution: A review,, North American Actuarial Journal, 13 (2009), 217.  doi: 10.1080/10920277.2009.10597549.  Google Scholar

[4]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency,, Insurance: Mathematics and Economics, 52 (2013), 98.  doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

[5]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion,, ASTIN Bulletin, 38 (2008), 653.  doi: 10.2143/AST.38.2.2033357.  Google Scholar

[6]

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

[7]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion,, ASTIN Bulletin, 41 (2011), 611.  doi: 10.2139/ssrn.1709174.  Google Scholar

[8]

B. Avanzi, V. Tu and B. Wong, On optimal periodic dividend strategies in the dual model with diffusion,, Insurance: Mathematics and Economics, 55 (2014), 210.  doi: 10.1016/j.insmatheco.2014.01.005.  Google Scholar

[9]

P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model,, Mathematical Finance, 15 (2005), 261.  doi: 10.1111/j.0960-1627.2005.00220.x.  Google Scholar

[10]

E. Bayraktar and M. Egami, Optimizing venture capital investments in a jump diffusion model,, Mathematical Methods of Operations Research, 67 (2008), 21.  doi: 10.1007/s00186-007-0181-x.  Google Scholar

[11]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model,, ASTIN Bulletin, 43 (2013), 359.  doi: 10.1017/asb.2013.17.  Google Scholar

[12]

E. Bayraktar, A. E. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs,, Insurance: Mathematics and Economics, 54 (2014), 133.  doi: 10.1016/j.insmatheco.2013.11.007.  Google Scholar

[13]

E. C. K. Cheung and S. Drekic, Dividend moments in the dual model: Exact and approximate approaches,, ASTIN Bulletin, 38 (2008), 399.  doi: 10.2143/AST.38.2.2033347.  Google Scholar

[14]

H. Dai, Z. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections,, Mathematical Methods of Operations Research, 72 (2010), 129.  doi: 10.1007/s00186-010-0312-7.  Google Scholar

[15]

H. Dai, Z. Liu and N. Luan, Optimal financing and dividend control in the dual model,, Mathematical and Computer Modelling, 53 (2011), 1921.  doi: 10.1016/j.mcm.2011.01.019.  Google Scholar

[16]

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

[17]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Applications of Mathematics, (1993).   Google Scholar

[18]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs,, Insurance: Mathematics and Economics, 44 (2009), 88.  doi: 10.1016/j.insmatheco.2008.10.001.  Google Scholar

[19]

S. Jaschke, A note on the inhomogeneous linear stochastic differential equation,, Insurance: Mathematics and Economics, 32 (2003), 461.  doi: 10.1016/S0167-6687(03)00134-3.  Google Scholar

[20]

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

[21]

K. Miyasawa, An economic survival game,, Journal of the Operations Research Society of Japan, 4 (1962), 95.   Google Scholar

[22]

H. Schmidli, Stochastic Control in Insurance,, Springer, (2008).   Google Scholar

[23]

D. J. Yao, H. L. Yang and R. M. Wang, Optimal financing and dividend strategies in a dual model with proportional costs,, Journal of Industrial and Management Optimization, 6 (2010), 761.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[24]

D. J. Yao, H. L. Yang and R. W. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs,, European Journal of Operational Research, 211 (2011), 568.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

[25]

D. J. Yao, R. W. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model,, Journal of Industrial and Management Optimization, 10 (2014), 1235.  doi: 10.3934/jimo.2014.10.1235.  Google Scholar

[26]

C. C. Yin and Y. Z. Wen, Optimal dividends problem with a terminal value for spectrally positive Lévy processes,, Insurance: Mathematics and Economics, 53 (2013), 769.  doi: 10.1016/j.insmatheco.2013.09.019.  Google Scholar

[27]

C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments,, Insurance: Mathematics and Economics, 52 (2013), 469.  doi: 10.1016/j.insmatheco.2013.02.014.  Google Scholar

[28]

C. C. Yin, Y. Z. Wen and Y. X. Zhao, On the optimal dividend problem for a spectrally positive Lévy process,, ASTIN Bulletin, 44 (2014), 635.  doi: 10.1017/asb.2014.12.  Google Scholar

[29]

Z. M. Zhang, On a risk model with randomized dividend-decision times,, Journal of Industrial and Management Optimization, 10 (2014), 1041.  doi: 10.3934/jimo.2014.10.1041.  Google Scholar

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