Optimality of Hybrid Continuous and Periodic Barrier Strategies in the Dual Model

  • José-Luis Pérez
  • Kazutoshi Yamazaki


Avanzi et al. (ASTIN Bull 46(3): 709–746, 2016) recently studied an optimal dividend problem where dividends are paid both periodically and continuously with different transaction costs. In the Brownian model with Poissonian periodic dividend payment opportunities, they showed that the optimal strategy is either of the pure-continuous, pure-periodic, or hybrid-barrier type. In this paper, we generalize the results of their previous study to the dual (spectrally positive Lévy) model. The optimal strategy is again of the hybrid-barrier type and can be concisely expressed using the scale function. These results are confirmed through a sequence of numerical experiments.


Dividends Lévy processes Periodic strategies Scale functions Dual model 

Mathematics Subject Classification

60G51 93E20 91B30 


  1. 1.
    Asmussen, S., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109(1), 79–111 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Avanzi, B., Tu, V., Wong, B.: On optimal periodic dividend strategies in the dual model with diffusion. Insur. Math. Econ. 55, 210–224 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avanzi, B., Tu, V., Wong, B.: On the interface between optimal periodic and continuous strategies in the presence of transaction costs. ASTIN Bull. 46(3), 709–746 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156–180 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avram, F., Pérez, J.L., Yamazaki, K.: Spectrally negative Lévy processes with Parisian reflection below and classical reflection above. Stoch. Process. Appl. 128(1), 255–290 (2018)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bayraktar, E., Kyprianou, A.E., Yamazaki, K.: On optimal dividends in the dual model. ASTIN Bull. 43(3), 359–372 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, T., Kyprianou, A.E., Savov, M.: Smoothness of scale functions for spectrally negative Lévy processes. Probab. Theory Relat. Fields 150, 691–708 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models. Stochastics 85(1), 111–143 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Egami, M., Yamazaki, K.: On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Lévy models. Adv. Appl. Probab. 46(1), 139–167 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Egami, M., Yamazaki, K.: Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hernández-Hernández, D., Pérez, J.L., Yamazaki, K.: Optimality of refraction strategies for spectrally negative Lévy processes. SIAM J. Control Optim. 54(3), 1126–1156 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuznetsov, A., Kyprianou, A.E., Rivero, V.: Lévy Matters II, Springer Lecture Notes in Mathematics. The theory of scale functions for spectrally negative Lévy processes. Springer, Berlin (2013)Google Scholar
  13. 13.
    Leung, T., Yamazaki, K., Zhang, H.: An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting. Int. J. Theor. Appl. Financ. 18(5), 1550032 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Loeffen, R.: 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Loeffen, R.L., Renaud, J.-F., Zhou, X.: Occupation times of intervals until first passage times for spectrally negative Lévy processes with applications. Stoch. Process. Appl. 124(3), 1408–1435 (2014)CrossRefzbMATHGoogle Scholar
  16. 16.
    Pérez, J.L., Yamazaki, K.: Mixed periodic-classical barrier strategies for Lévy risk process (2016). arXiv:1609.01671
  17. 17.
    Pérez, J.L., Yamazaki, K.: Hybrid continuous and periodic barrier strategies in the dual model: optimality and fluctuation identities (2016). arXiv:1612.02444
  18. 18.
    Pérez, J.L., Yamazaki, K.: On the optimality of periodic barrier strategies for a spectrally positive Lévy process. Insur. Math. Econ. 77, 1–13 (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pérez, J.L., Yamazaki, K.: On the refracted-reflected spectrally negative Lévy processes. Stoch. Process. Appl. 128(1), 306–331 (2018)CrossRefzbMATHGoogle Scholar
  20. 20.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005)CrossRefGoogle Scholar
  21. 21.
    Yamazaki, K.: Contraction options and optimal multiple-stopping in spectrally negative Lévy models. Appl. Math. Optim. 72(1), 147–185 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yamazaki, K.: Inventory control for spectrally positive Lévy demand processes. Math. Oper. Res. 42(1), 212–237 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Probability and StatisticsCentro de Investigación en MatemáticasGuanajuatoMexico
  2. 2.Department of Mathematics, Faculty of Engineering ScienceKansai UniversitySuita-shiJapan

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