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Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls

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  • Zhuo Jin

    (The University of Melbourne)

  • G. Yin

    (Wayne State University)

Abstract

This work focuses on numerical methods for finding optimal dividend payment and investment policies to maximize the present value of the cumulative dividend payment until ruin; the surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the optimal value function obeys a coupled system of nonlinear integro-differential quasi-variational inequalities. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to approximate the value function and optimal controls. Convergence of the approximation algorithms are proved. Examples are presented to illustrate the applicability of the numerical methods.

Suggested Citation

  • Zhuo Jin & G. Yin, 2013. "Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 246-271, October.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:1:d:10.1007_s10957-012-0263-7
    DOI: 10.1007/s10957-012-0263-7
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    References listed on IDEAS

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    1. Hans U. Gerber, 1972. "Games of Economic Survival with Discrete- and Continuous-Income Processes," Operations Research, INFORMS, vol. 20(1), pages 37-45, February.
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    3. Asmussen, Soren & Taksar, Michael, 1997. "Controlled diffusion models for optimal dividend pay-out," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 1-15, June.
    4. Pablo Azcue & Nora Muler, 2010. "Optimal investment policy and dividend payment strategy in an insurance company," Papers 1010.4988, arXiv.org.
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    Citations

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    Cited by:

    1. Yongwu Li & Zhongfei Li & Yan Zeng, 2016. "Equilibrium Dividend Strategy with Non-exponential Discounting in a Dual Model," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 699-722, February.
    2. Zhang, Nan & Jin, Zhuo & Qian, Linyi & Fan, Kun, 2019. "Stochastic differential reinsurance games with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 7-18.
    3. Zhuo Jin & Zuo Quan Xu & Bin Zou, 2020. "A Perturbation Approach to Optimal Investment, Liability Ratio, and Dividend Strategies," Papers 2012.06703, arXiv.org, revised May 2021.
    4. Kei Noba & Jos'e-Luis P'erez & Xiang Yu, 2019. "On the bail-out dividend problem for spectrally negative Markov additive models," Papers 1901.03021, arXiv.org, revised Feb 2020.
    5. Hongjiang Qian & Zhexin Wen & George Yin, 2022. "Numerical solutions for optimal control of stochastic Kolmogorov systems with regime-switching and random jumps," Statistical Inference for Stochastic Processes, Springer, vol. 25(1), pages 105-125, April.
    6. Yan Wang & Lei Wang & Kok Lay Teo, 2018. "Necessary and Sufficient Optimality Conditions for Regular–Singular Stochastic Differential Games with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 501-532, November.
    7. Zhuo Jin, 2015. "Optimal Debt Ratio and Consumption Strategies in Financial Crisis," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 1029-1050, September.
    8. Etienne Chevalier & Vathana Ly Vath & Alexandre Roch, 2020. "Optimal Dividend and Capital Structure with Debt Covenants," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 535-565, November.
    9. Y. Zhang & Z. Jin & J. Wei & G. Yin, 2022. "Mean-variance portfolio selection with dynamic attention behavior in a hidden Markov model," Papers 2205.08743, arXiv.org.
    10. Xixi Yang & Jiyang Tan & Hanjun Zhang & Ziqiang Li, 2017. "An Optimal Control Problem in a Risk Model with Stochastic Premiums and Periodic Dividend Payments," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(03), pages 1-18, June.

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