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Journal of Optimization Theory and Applications

, Volume 179, Issue 2, pp 501–532 | Cite as

Necessary and Sufficient Optimality Conditions for Regular–Singular Stochastic Differential Games with Asymmetric Information

  • Yan Wang
  • Lei WangEmail author
  • Kok Lay Teo
Article
  • 230 Downloads

Abstract

We consider a class of regular–singular stochastic differential games arising in the optimal investment and dividend problem of an insurer under model uncertainty. The information available to the two players is asymmetric partial information and the control variable of each player consists of two components: regular control and singular control. We establish the necessary and sufficient optimality conditions for the saddle point of the zero-sum game. Then, as an application, these conditions are applied to an optimal investment and dividend problem of an insurer under model uncertainty. Furthermore, we generalize our results to the nonzero-sum regular–singular game with asymmetric information, and then the Nash equilibrium point is characterized.

Keywords

Necessary optimality conditions Sufficient optimality conditions Regular–singular control Stochastic differential game Asymmetric information Saddle point Nash equilibrium Optimal investment Dividend Model uncertainty 

Mathematics Subject Classification

91G80 91A15 91A23 93E20 60J75 91B28 91B30 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation for the Youth of China (Grants 11701064, 11301081, 11401073), the Science Research Project of Educational Department of Liaoning Province of China (Grants L2014188, L2015097 and L2014186), the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25), and the Research Funding for Doctor Start-Up Program of Liaoning Province (Grant 201601245).

References

  1. 1.
    Markowitz, H.: Portfolio selection*. J. Finance 7, 77–91 (1952)Google Scholar
  2. 2.
    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)CrossRefGoogle Scholar
  3. 3.
    Højgaard, B., Taksar, M.: Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quant. Finance 4, 315–327 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Azcue, P., Muler, N.: Optimal investment policy and dividend payment strategy in an insurance company. Ann. Appl. Probab. 20, 1253–1302 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jin, Z., Yin, G.: Numerical methods for optimal dividend payment and investment strategies of Markov-modulated jump diffusion models with regular and singular controls. J. Optim. Theory Appl. 159, 246–271 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wang, Y., Zhao, Y., Wang, L., Song, A., Ma, Y.: Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. J. Ind. Manag. Optim.  https://doi.org/10.3934/jimo.2017067 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hansen, L.P., Sargent, T.J.: Robust control and model uncertainty. Amer. Econ. Rev. 91, 60–66 (2001)CrossRefGoogle Scholar
  8. 8.
    Cont, R.: Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance 16, 519–547 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhang, X., Siu, T.K.: Optimal investment and reinsurance of an insurer with model uncertainty. Insur. Math. Econ. 45, 81–88 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Elliott, R.J., Siu, T.K.: A stochastic differential game for optimal investment of an insurer with regime switching. Quant. Finance 11, 365–380 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Peng, X., Chen, F., Hu, Y.: Optimal investment, consumption and proportional reinsurance under model uncertainty. Insur. Math. Econ. 59, 222–234 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wang, Y., Wang, L.: Forward-backward stochastic differential games for optimal investment and dividend problem of an insurer under model uncertainty. Appl. Math. Model.  https://doi.org/10.1016/j.apm.2017.07.027 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    An, T.T.K., Øksendal, B.: Maximum principle for stochastic differential games with partial information. J. Optim. Theory Appl. 139, 463–483 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Øksendal, B., Sulèm, A.: Singular stochastic control and optimal stopping with partial information of Itô-Lévy processes. SIAM J. Control Optim. 50, 2254–2287 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, Y., Song, A., Feng, E.: A maximum principle via Malliavin calculus for combined stochastic control and impulse control of forward-backward systems. Asian J. Control 17, 1798–1809 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Øksendal, B., Sulèm, A.: Applied Stochastic Control of Jump Diffusions. Springer, Berlin (2007)CrossRefGoogle Scholar
  18. 18.
    Cadenillas, A., Haussmann, U.G.: The stochastic maximum principle for a singular control problem. Stochastics 49, 211–237 (1994)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dufour, F., Miller, B.: Maximum principle for singular stochastic control problems. SIAM J. Control Optim. 45, 668–698 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bahlali, S., Djehiche, B., Mezerdi, B.: The relaxed stochastic maximum principle in singular optimal control of diffusions. SIAM J. Control Optim. 46, 427–444 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Andersson, D.: The relaxed general maximum principle for singular optimal control of diffusions. Syst. Control Lett. 58, 76–82 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hafayed, M., Abbas, S.: On near-optimal mean-field stochastic singular controls: necessary and sufficient conditions for near-optimality. J. Optim. Theory Appl. 160, 778–808 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, R., Liu, B.: Necessary and sufficient near-optimal conditions for mean-field singular stochastic controls. Asian J. Control 17, 1209–1221 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, F.: Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems. J. Syst. Sci. Complex. 26, 886–901 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hafayed, M., Abba, A., Abbas, S.: On partial-information optimal singular control problem for mean-field stochastic differential equations driven by Teugels martingales measures. Int. J. Control 89, 397–410 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hu, Y., Øksendal, B., Sulèm, A.: Singular mean-field control games with applications to optimal harvesting and investment problems. arXiv:1406.1863, (2014)
  27. 27.
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, New York (2009)CrossRefGoogle Scholar
  28. 28.
    Dufresne, F., Gerber, H.U.: Risk theory for the compound Poisson process that is perturbed by diffusion. Insur. Math. Econ. 10, 51–59 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Shi, J.T., Wang, G.C., Xiong, J.: Leader-follower stochastic differential game with asymmetric information and applications. Automatica 63, 60–73 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceDalian Jiaotong UniversityDalianChina
  2. 2.School of Mathematical SciencesDalian University of TechnologyDalianChina
  3. 3.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

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