Skip to main content
SpringerLink
Log in

Equilibrium dividend strategies in the dual model with a random time horizon

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

This paper investigates the dividend problem with non-exponential discounting in a dual model. We assume that the dividends can only be paid at a bounded rate and that the surplus process is killed by an exponential random variable. Since the non-exponential discount function leads to a time inconsistent control problem, we study the equilibrium HJB-equation and give the associated verification theorem. For the case of a mixture of exponential discount functions and exponential gains, we obtain the explicit equilibrium dividend strategy and the corresponding equilibrium value function. Besides, numerical examples are shown to illustrate our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H Albrecher, S Thonhauser. On optimal dividend strategies in insurance with a random time horizon, Stochastic Processes, Finance and Control: A Festschrift in Honor of Robert J Elliott, 2012, 157–179.

  2. B Avanzi, H U Gerber, E S Shiu. Optimal dividends in the dual model, Insurance Math Econom, 2007, 41(1): 111–123.

    Article  MathSciNet  Google Scholar 

  3. B Avanzi, J Shen, B Wong. Optimal dividends and capital injections in the dual model with diffusion, Astin Bull, 2011, 41(2): 611–644.

    MathSciNet  Google Scholar 

  4. T Björk, A Murgoci. A general theory of Markovian time inconsistent stochastic control problems, Working paper, Stockholm School of Economics, 2010.

  5. T Björk, A Murgoci, X Y Zhou. Mean-Variance Portfolio Optimization with State-Dependent Risk Aversion, Math Finance, 2014, 24(1): 1–24.

    Article  MathSciNet  Google Scholar 

  6. S Chen, Z Li, Y Zeng. Optimal dividend strategies with time-inconsistent preferences, J Econom Dynam Control, 2014, 46: 150–172.

    Article  MathSciNet  Google Scholar 

  7. S Chen, X Wang, Y Deng, Y Zeng. Optimal dividend-financing strategies in a dual risk model with time-inconsistent preferences, Insurance Math Econom, 2016, 67: 27–37.

    Article  MathSciNet  Google Scholar 

  8. B de Finetti. Su una impostazione alternativa della teoria collettiva del rischio, Transaction XVth International Congress of actuaries, 1957, 2: 433–443.

    Google Scholar 

  9. I Ekeland, A Lazrak. Being serious about non-commitment: subgame perfect equilibrium in continuous time, 2006, arXiv preprint math/0604264.

  10. I Ekeland, O Mbodji, T A Pirvu. Time-consistent portfolio management, SIAM J Financial Math, 2012, 3(1): 1–32.

    Article  MathSciNet  Google Scholar 

  11. H U Gerber, E S Shiu. On optimal dividend strategies in the compound Poisson model, N Am Actuar J, 2006, 10(2): 76–93.

    Article  MathSciNet  Google Scholar 

  12. Y Li, Z Li, Y Zeng. Equilibrium Dividend Strategy with Non-exponential Discounting in a Dual Model, J Optim Theory Appl, 2016, 168(2): 699–722.

    Article  MathSciNet  Google Scholar 

  13. R H Strotz. Myopia and inconsistency in dynamic utility maximization, Rev Econom Stud, 1955, 23(3): 165–180.

    Article  Google Scholar 

  14. S Thonhauser, H Albrecher. Dividend maximization under consideration of the time value of ruin, Insurance Math Econom, 2007, 41(1): 163–184.

    Article  MathSciNet  Google Scholar 

  15. J Yong. Time-inconsistent optimal control problems and the equilibrium HJB-equation, Math Control Relat F, 2012, 2(3): 271–329.

    Article  MathSciNet  Google Scholar 

  16. Y Zeng, Z Li. Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math Econom, 2011, 49(1): 145–154.

    Article  MathSciNet  Google Scholar 

  17. Q Zhao, J Wei, R Wang. On dividend strategies with non-exponential discounting, Insurance Math Econom, 2014, 58: 1–13.

    Article  MathSciNet  Google Scholar 

  18. Y Zhao, H Dong, W Zhong. Equilibrium dividend strategies for spectrally negative Lvy processes with time value of ruin and random time horizon, Communications in Statistics-Theory and Methods, 2020, 1–24.

  19. Y Zhao, P Chen, H Yang. Optimal periodic dividend and capital injection problem for spectrally positive Lvy processes, Insurance Math Econom, 2017, 74: 135–146.

    Article  MathSciNet  Google Scholar 

  20. Y Zhao, R Wang, D Yao, P Chen. Optimal dividends and capital injections in the dual model with a random time horizon, J Optim Theory Appl, 2015, 167(1): 272–295.

    Article  MathSciNet  Google Scholar 

  21. Y Zhao, D Yao. Optimal dividend and capital injection problem with a random time horizon and a ruin penalty in the dual model, Appl Math J Chinese Univ, 2015, 30(3): 325–339.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Statistics and Data Science, Qufu Normal University, Qufu, 273165, China

    Yong-xia Zhao

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Chuan-xiu Ye

  3. School of Economics, Nanjing University of Finance and Economics, Nanjing, 210046, China

    Gong-pin Cheng

Authors
  1. Yong-xia Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuan-xiu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gong-pin Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yong-xia Zhao, Chuan-xiu Ye or Gong-pin Cheng.

Ethics declarations

Conflict of interest The authors declare no conflict of interest.

Additional information

Supported by the Shandong Provincial Natural Science Foundation of China(ZR2020MA035 and ZR2023MA093).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, Yx., Ye, Cx. & Cheng, Gp. Equilibrium dividend strategies in the dual model with a random time horizon. Appl. Math. J. Chin. Univ. 38, 510–522 (2023). https://doi.org/10.1007/s11766-023-3751-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-023-3751-7

Keywords

MR Subject Classification

Search

Navigation