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

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  • Chuancun Yin
  • Kam Chuen Yuen

Abstract

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.

Suggested Citation

  • Chuancun Yin & Kam Chuen Yuen, 2014. "Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs," Papers 1409.0407, arXiv.org.
  • Handle: RePEc:arx:papers:1409.0407
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    File URL: http://arxiv.org/pdf/1409.0407
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    References listed on IDEAS

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    1. Avanzi, Benjamin & U. Gerber, Hans & S.W. Shiu, Elias, 2007. "Optimal dividends in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 111-123, July.
    2. He, Lin & Liang, Zongxia, 2009. "Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 88-94, February.
    3. Chuancun Yin & Yuzhen Wen, 2013. "Optimal dividends problem with a terminal value for spectrally positive Levy processes," Papers 1302.6011, arXiv.org.
    4. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2013. "On Optimal Dividends In The Dual Model," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 43(03), pages 359-372, September.
    5. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    6. repec:spr:compst:v:67:y:2008:i:1:p:21-42 is not listed on IDEAS
    7. Gerber, Hans U. & Smith, Nathaniel, 2008. "Optimal dividends with incomplete information in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 227-233, October.
    8. Kulenko, Natalie & Schmidli, Hanspeter, 2008. "Optimal dividend strategies in a Cramér-Lundberg model with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 270-278, October.
    9. Erhan Bayraktar & Masahiko Egami, 2008. "Optimizing venture capital investments in a jump diffusion model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 21-42, February.
    10. Chuancun Yin & Yuzhen Wen, 2013. "An extension of Paulsen-Gjessing's risk model with stochastic return on investments," Papers 1302.6757, arXiv.org.
    11. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2011. "Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs," European Journal of Operational Research, Elsevier, vol. 211(3), pages 568-576, June.
    12. Avanzi, Benjamin & Tu, Vincent & Wong, Bernard, 2014. "On optimal periodic dividend strategies in the dual model with diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 210-224.
    13. Yin, Chuancun & Wen, Yuzhen, 2013. "An extension of Paulsen–Gjessing’s risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 469-476.
    14. Yin, Chuancun & Wen, Yuzhen, 2013. "Optimal dividend problem with a terminal value for spectrally positive Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 769-773.
    15. repec:spr:compst:v:72:y:2010:i:1:p:129-143 is not listed on IDEAS
    16. Avanzi, Benjamin & Gerber, Hans U., 2008. "Optimal Dividends in the Dual Model with Diffusion," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 38(02), pages 653-667, November.
    17. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér-Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308.
    18. Jaschke, Stefan, 2003. "A note on the inhomogeneous linear stochastic differential equation," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 461-464, July.
    19. Avanzi, Benjamin & Shen, Jonathan & Wong, Bernard, 2011. "Optimal Dividends and Capital Injections in the Dual Model with Diffusion," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 41(02), pages 611-644, November.
    20. Chuancun Yin & Yuzhen Wen & Yongxia Zhao, 2013. "On the optimal dividend problem for a spectrally positive Levy process," Papers 1302.2231, arXiv.org, revised Mar 2014.
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