Skip to main content Accessibility help

On optimal periodic dividend and capital injection strategies for spectrally negative Lévy models

  • Kei Noba (a1), José-Luis Pérez (a2), Kazutoshi Yamazaki (a3) and Kouji Yano (a1)


De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.


Corresponding author

* Postal address: Department of Mathematics, Graduate School of Science, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan.
** Email address:
*** Postal address: Department of Probability and Statistics, Centro de Investigación en Matemáticas, A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico. Email address:
**** Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680, Japan. Email address:
***** Email address:


Hide All
[1]Albrecher, H., Bäuerle, N. and Thonhauser, S. (2011). Optimal dividend-payout in random discrete time. Statist. Risk Model. 28, 251276.
[2]Albrecher, H., Cheung, E. C. K. and Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model dividends. ASTIN Bull. 41, 645672.
[3]Avanzi, B., Tu, V. and Wong, B. (2014). On optimal periodic dividend strategies in the dual model with diffusion. Insurance Math. Econom. 55, 210224.
[4]Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156180.
[5]Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2014). Optimal dividends in the dual model under transaction costs. Insurance Math. Econom. 54, 133143.
[6]Chan, T., Kyprianou, A. E. and Savov, M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Relat. Fields 150, 691708.
[7]Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 122.
[8]Egami, M. and Yamazaki, K. (2013). Precautionary measures for credit risk management in jump models. Stochastsics 85, 111143.
[9]Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2013). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II, (Lecture Notes Math. 2061). Springer, Heidelberg, pp. 97186.
[10]Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.
[11]Leung, T., Yamazaki, K. and Zhang, H. (2015).An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting. Int. J. Theor. Appl. Finance 18, 1550032.
[12]Loeffen, R. L., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative L\'evy processes with applications. Stochastic Process. Appl. 124, 14081435.
[13]Maier, R. S. and O'Cinneide, C. A. (1992). A closure characterisation of phase-type distributions. J. Appl. Prob. 29, 92103.
[14]Noba, K., Pérez, J.-L., Yamazaki, K. and Yano, K. (2018). On optimal periodic dividend strategies for Lévy risk processes. Insurance Math. Econom. 80, 2944.
[15]Pérez, J.-L. and Yamazaki, K. (2017). On the optimality of periodic barrier strategies for a spectrally positive Lévy processes. Insurance Math. Econom. 77, 113.
[16]Pérez, J.-L. and Yamazaki, K. (2018). Mixed periodic-classical barrier strategies for Lévy risk processes. Risks 6, 39.
[17]Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. (Stoch. Model. Appl. Prob. 21). Springer, Berlin.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed