Skip to main content
SpringerLink
Log in

Optimal Singular Dividend Problem Under the Sparre Andersen Model

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the optimal dividend problem assuming that the underlying reserve process follows the Sparre Andersen model. In this model, there is no constant restriction on the dividend rates, i.e., the optimization problem is of singular type. In this case, the value function is no longer bounded and the associated Hamilton–Jacobi–Bellman equation is a variational inequality involving a first-order integro-differential operator and a gradient constraint. We prove the regularity properties for the value function by constructing strategies and show that the value function is a constrained viscosity solution of the associated Hamilton–Jacobi–Bellman equation. In addition, we prove that the value function is the upper semicontinuous envelope of the supremum for a class of subsolutions.

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. De Finetti, B., Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2, pp. 433–443. New York (1957)

  2. Asmussen, S., Taksar, M.: Controlled diffusion models for optimal dividend pay-out. Insur. Math. Econ. 20(1), 1–15 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, X., Liu, J., Zhou, X.Y.: A constrained non-linear regular-singular stochastic control problem, with applications. Stoch. Process. Appl. 109(2), 167–187 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Azcue, P., Muler, N.: Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15(2), 261–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belhaj, M.: Optimal dividend payments when cash reserves follow a jump-diffusion process. Math. Finance 20(2), 313–325 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Andersen, E.S.: On the collective theory of risk in case of contagion between claims. Bull. Inst. Math. Appl. 12(2), 275–279 (1957)

    Google Scholar 

  8. Li, S.M., Garrido, J.: On a class of renewal risk models with a constant dividend barrier. Insur. Math. Econ. 35(3), 691–701 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Albrecher, H., Mármol, M., Claramunt, M.M.: On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times. Insur. Math. Econ. 37(2), 324–334 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Albrecher, H., Hartinger, J.: On the non-optimality of horizontal barrier strategies in the Sparre Andersen model. HERMES Int. J. Comput. Math. Appl. 7, 1–14 (2006)

    MATH  Google Scholar 

  11. Albrecher, H., Thonhauser, S.: Optimality results for dividend problems in insurance. RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 103(2), 295–320 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Azcue, P., Muler, N.: Optimal investment policy and dividend payment strategy in an insurance company. Ann. Appl. Probab. 20(4), 1253–1302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benth, F.E., Karlsen, K.H., Reikvam, K.: Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach. Finance Stoch. 5(3), 275–303 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Seydel, R.C.: Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. Stoch. Process. Appl 119(10), 3719–3748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lions, P.L.: Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations part 2: viscosity solutions and uniqueness. Commun. Partial Differ. Equ. 8(11), 1229–1276 (1983)

    Article  MATH  Google Scholar 

  18. Lions, P.L.: Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. I. The dynamic programming principle and applications. Commun. Partial Differ. Equ. 8(10), 1101–1174 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110–1122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  21. Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43. Springer, New York (1999)

    Book  MATH  Google Scholar 

  22. Bai, L.H., Ma, J., Xing, X.J.: Optimal dividend and investment problems under Sparre Andersen model. Ann. Appl. Probab. 27(6), 3588–3632 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.L.: Stochastic Processes for Insurance and Finance, vol. 505. Wiley, New York (2009)

    MATH  Google Scholar 

  24. Protter, P.: Stochastic integration and differential equations (2nd ed.). In: Stochastic Modelling and Applied Probability, vol. 21. Springer-Verlag, Berlin (2004)

  25. Scheer, N., Schmidli, H.: Optimal dividend strategies in a Cramér–Lundberg model with capital injections and administration costs. Eur. Actuar. J. 1(1), 57–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  27. Awatif, S.: Equqtions d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels: Partie 1: Unicité Des solutions de viscosité. Commun. Partial Differ. Equ. 16(6–7), 1057–1074 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Czarna, I., Palmowski, Z.: Dividend problem with Parisian delay for a spectrally negative Lévy risk process. J. Optim. Theory Appl. 161(1), 239–256 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Marciniak, E., Palmowski, Z.: On the optimal dividend problem for insurance risk models with surplus-dependent premiums. J. Optim. Theory Appl. 168(2), 723–742 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pérez, J.L., Yamazaki, K., Yu, X.: On the bail-out optimal dividend problem. J. Optim. Theory Appl. 179(2), 553–568 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhu, J., Yang, H.: Optimal capital injection and dividend distribution for growth restricted diffusion models with bankruptcy. Insur. Math. Econ. 70, 259–271 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhu, J., Chen, F.: Dividend optimization for regime-switching general diffusions. Insur. Math. Econ. 53(2), 439–456 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Wei, J., Yang, H., Wang, R.: Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. J. Optim.Theory Appl. 147(2), 358–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Choulli, T., Taksar, M., Zhou, X.Y.: A diffusion model for optimal dividend distribution for a company with constraints on risk control. SIAM J. Control Optim. 41(6), 1946–1979 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mou, C.: Perron’s method for nonlocal fully nonlinear equations. Anal. PDE 10(5), 1227–1254 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Caffarelli, L.A., Ros-Oton, X., Serra, J.: Obstacle problems for integro-differential operators: regularity of solutions and free boundaries. Invent. Math. 208(3), 1155–1211 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Here, we want to express our thanks to Jacques Rioux for his dedication to the improvement in this paper. This research is supported by Chinese NSF Grants No. 11471171, No. 11911530091 and No. 11931018.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, China

    Linlin Tian, Lihua Bai & Junyi Guo

Authors
  1. Linlin Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lihua Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junyi Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lihua Bai.

Additional information

Kok Lay Teo.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, L., Bai, L. & Guo, J. Optimal Singular Dividend Problem Under the Sparre Andersen Model. J Optim Theory Appl 184, 603–626 (2020). https://doi.org/10.1007/s10957-019-01600-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01600-0

Keywords

Mathematics Subject Classification

Search

Navigation