Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-28T16:51:53.689Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  16 January 2019

Kei Noba*
Affiliation:
Kyoto University
José-Luis Pérez*
Affiliation:
Centro de Investigación en Matemáticas
Kazutoshi Yamazaki*
Affiliation:
Kansai University
Kouji Yano*
Affiliation:
Kyoto University
*
* Postal address: Department of Mathematics, Graduate School of Science, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan.
*** 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: jluis.garmendia@cimat.mx
**** Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680, Japan. Email address: kyamazak@kansai-u.ac.jp
* Postal address: Department of Mathematics, Graduate School of Science, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan.

Abstract

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albrecher, H., Bäuerle, N. and Thonhauser, S. (2011). Optimal dividend-payout in random discrete time. Statist. Risk Model. 28, 251276.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[5]Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2014). Optimal dividends in the dual model under transaction costs. Insurance Math. Econom. 54, 133143.Google Scholar
[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.Google Scholar
[7]Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 122.Google Scholar
[8]Egami, M. and Yamazaki, K. (2013). Precautionary measures for credit risk management in jump models. Stochastsics 85, 111143.Google Scholar
[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.Google Scholar
[10]Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.Google Scholar
[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. Google Scholar
[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.Google Scholar
[13]Maier, R. S. and O'Cinneide, C. A. (1992). A closure characterisation of phase-type distributions. J. Appl. Prob. 29, 92103.Google Scholar
[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.Google Scholar
[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.Google Scholar
[16]Pérez, J.-L. and Yamazaki, K. (2018). Mixed periodic-classical barrier strategies for Lévy risk processes. Risks 6, 39.Google Scholar
[17]Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. (Stoch. Model. Appl. Prob. 21). Springer, Berlin.Google Scholar