Optimal dividends with an affine penalty

  • Zhibin LiangEmail author
  • Virginia R. Young
Original Research


We find the optimal dividend strategy in two related risk models under an affine penalty for ruin. The first risk model is the classical Cramér–Lundberg risk model, and the second is the so-called dual risk model. Under both models, for exponentially distributed jumps, we show that the optimal dividend strategy is a barrier strategy and obtain the barrier explicitly. Moreover, we prove that the optimal barrier increases with respect to the parameters of the affine penalty, while the penalized value function decreases with respect to the penalty. We also compute the expected time until ruin and show that the expected time of ruin increases with respect to the parameters of the affine penalty. Finally, we present some numerical examples to demonstrate the relationship between the results for the classical and dual risk models. Our main contributions are in comparing the classical and dual risk models side-by-side and in obtaining explicit expressions for the penalized value functions, the optimal barriers, and the expected times of ruin. Also, we contrast the free-boundary condition associated with the barrier strategies in the two models.


Optimal dividend strategy Ruin penalty Classical risk model Dual risk model Instantaneous control Impulse control 

Mathematics Subject Classification

93E20 91B30 

JEL Classification

C58 C61 G22 



The authors would like to thank the anonymous referees for their careful reading and helpful comments on an earlier version of this paper, which led to a considerable improvement of the presentation of the work.


  1. 1.
    Avanzi, Benjamin: Strategies for dividend distribution: a review. N. Am. Actuar. J. 13(2), 217–251 (2009)MathSciNetGoogle Scholar
  2. 2.
    Avanzi, Benjamin, Gerber, Hans U.: Optimal dividends in the dual model with diffusion. ASTIN Bull. 38(2), 653–667 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Avanzi, Benjamin, Gerber, Hans U., Shiu, Elias S.W.: Optimal dividends in the dual model. Insur. Math. Econ. 41(1), 111–123 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Avram, Florin, Palmowski, Zbigniew, Pistorius, Martijn R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17(1), 156–180 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Avram, Florin, Palmowski, Zbigniew, Pistorius, Martijn R.: On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk-process in the presence of a penalty function. Ann. Appl. Probab. 25(4), 1868–1935 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Albrecher, Hansjörg, Thonhauser, Stefan: Optimal dividend strategies for a risk process under force of interest. Insur. Math. Econ. 43(1), 134–149 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hansjörg, Albrecher, Thonhauser, Stefan: 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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Azcue, Pablo, Muler, Nora: Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15(2), 261–308 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bayraktar, Erhan, Kyprianou, Andreas E., Yamazaki, Kazutoshi: Optimal dividends in the dual model under transaction costs. Insur. Math. Econ. 54, 133–143 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bühlmann, Hans: Mathematical Methods in Risk Theory. Springer, Heidelberg (1970)zbMATHGoogle Scholar
  11. 11.
    De Finetti, B.: Su unimpostazione alternativa della teoria collettiva del rischio. Trans. XVth Int. Congr. Actuar. 2(1), 433–443 (1957)Google Scholar
  12. 12.
    Dickson, David C.M., Waters, Howard R.: Some optimal dividends problems. ASTIN Bull. 34(1), 49–74 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dixit, Avinash: A simplified treatment of the theory of optimal regulation of Brownian motion. J. Econ. Dyn. Control 15(4), 657–673 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dufresne, Daniel: Fitting combinations of exponentials to probability distributions. Appl. Stoch. Models Bus. Ind. 23(1), 23–48 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dumas, Bernard: Super contact and related optimality conditions. J. Econ. Dyn. Control 15(4), 675–685 (1991)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gerber, Hans U., Sheldon Lin, X., Yang, Hailiang: A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bull. 36(2), 489–503 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gerber, Hans U., Shiu, Elias S.W., Smith, Nathaniel: Maximizing dividends without bankruptcy. ASTIN Bull. 36(1), 5–23 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kulenko, Natalie, Schmidli, Hanspeter: Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insur. Math. Econ. 43(2), 270–278 (2008)zbMATHGoogle Scholar
  19. 19.
    Kyprianou, Andreas E., Palmowski, Zbigniew: Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44, 428–443 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Li, Yongwu, Li, Zhongfei, Zeng, Yan: Equilibrium dividend strategy with non-exponential discounting in a dual model. J. Optim. Theory Appl. 168(2), 699–722 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Liang, Zhibin, Young, Virginia R.: Dividends and reinsurance under a penalty for ruin. Insur. Math. Econ. 50(3), 437–445 (2012)zbMATHGoogle Scholar
  22. 22.
    Loeffen, Ronnie L.: On optimality of the barrier strategy in De Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18(5), 1669–1680 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Loeffen, Ronnie L.: An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. J. Appl. Probab. 46(1), 85–98 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Loeffen, Ronnie L.: An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insur. Math. Econ. 45(1), 41–48 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Loeffen, Ronnie L., Renaud, Jean-François: De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insur. Math. Econ. 46(1), 98–108 (2010)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Marciniak, Ewa, Palmowski, Zbigniew: On the optimal dividend problem for insurance risk models with surplus-dependent premiums. J. Optim. Theory Appl. 168(2), 723–742 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ng, Andrew C.Y.: On a dual model with a dividend threshold. Insur. Math. Econ. 44(2), 315–324 (2009)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Scheer, Natalie, Schmidli, Hanspeter: Optimal dividend strategies in a Cramér–Lundberg model with capital injections and administration costs. Eur. Actuar. J. 1(1), 57–92 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Schmidli, Hanspeter: On capital injections and dividends with tax in a classical risk model. Insur. Math. Econ. 71, 138–144 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Shang, Yilun: Deffuant model with general opinion distributions: first impression and critical confidence bound. Complexity 19(2), 38–49 (2013)MathSciNetGoogle Scholar
  31. 31.
    Shang, Yilun: An agent based model for opinion dynamics with random confidence threshold. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3766–3777 (2014)MathSciNetGoogle Scholar
  32. 32.
    Thonhauser, Stefan, Albrecher, Hansjörg: Dividend maximization under consideration of the time value of ruin. Insur. Math.Econ. 41(1), 163–184 (2007)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Vierkötter, Matthias, Schmidli, Hanspeter: On optimal dividends with exponential and linear penalty payments. Insur. Math. Econ. 72, 265–270 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yao, Dingjun, Yang, Hailiang, Wang, Rongming: Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. Eur. J. Oper. Res. 211(3), 568–576 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhou, Ming, Yiu, Ka Fai Cedric: Optimal dividend strategy with transaction costs for an upward jump model. Quant. Finance 14(6), 1097–1106 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesNanjing Normal UniversityJiangsuChina
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUnited States

Personalised recommendations