Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T09:21:55.711Z Has data issue: false hasContentIssue false
  • English
  • Français

REFRACTION–REFLECTION STRATEGIES IN THE DUAL MODEL

Published online by Cambridge University Press:  03 October 2016

José-Luis Pérez  and
Kazutoshi Yamazaki
José-Luis Pérez
Affiliation:
Department of Probability and Statistics, Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico, E-Mail: [email protected]
Kazutoshi Yamazaki*
Affiliation:
Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680, Japan
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the dual model with capital injection under the additional condition that the dividend strategy is absolutely continuous. We consider a refraction–reflection strategy that pays dividends at the maximal rate whenever the surplus is above a certain threshold, while capital is injected so that it stays non-negative. The resulting controlled surplus process becomes the spectrally positive version of the refracted–reflected process recently studied by Pérez and Yamazaki (2015). We study various fluctuation identities of this process and prove the optimality of the refraction–reflection strategy. Numerical results on the optimal dividend problem are also given.

Keywords

Dividendscapital injectionrefracted-reflected Lévy processesscale functionsdual model
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 1 , January 2017 , pp. 199 - 238
Copyright
Copyright © Astin Bulletin 2016 

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

Asmussen, S., Avram, F. and Pistorius, M.R. (2004) Russian and American put options under exponential phase-type Lévy models. Stochastic Processes and their Applications, 109 (1), 79–111.CrossRefGoogle Scholar
Avanzi, B. and Gerber, H.U. (2008) Optimal dividends in the dual model with diffusion. Astin Bulletin, 38 (2), 653–667.CrossRefGoogle Scholar
Avanzi, B., Gerber, H.U. and Shiu, E. (2007) Optimal dividends in the dual model. Insurance: Mathematics and Economics, 41 (1), 111–123.Google Scholar
Avanzi, B., Shen, J. and Wong, B. (2011) Optimal dividends and capital injections in the dual model with diffusion. Astin Bulletin, 41 (2), 611–644.Google Scholar
Avanzi, B., Tu, V. and Wong, B. (2014) On optimal periodic dividend strategies in the dual model with diffusion. Insurance: Mathematics and Economics, 55, 210–224.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M.R. (2007) On the optimal dividend problem for a spectrally negative Lévy process. Annals of Applied Probability, 17, 156–180.CrossRefGoogle Scholar
Bayraktar, E., Kyprianou, A.E. and Yamazaki, K. (2013) On optimal dividends in the dual model. Astin Bulletin, 43 (3), 359–372.CrossRefGoogle Scholar
Bayraktar, E., Kyprianou, A.E. and Yamazaki, K. (2014) Optimal dividends in the dual model under transaction costs. Insurance: Mathematics and Economics 54, 133–143.Google Scholar
Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press.Google Scholar
Chan, T., Kyprianou, A.E. and Savov, M. (2011) Smoothness of scale functions for spectrally negative Lévy processes. Probability Theory and Related Fields, 150, 691–708.CrossRefGoogle Scholar
Egami, M. and Yamazaki, K. (2013) Precautionary measures for credit risk management in jump models. Stochastics, 85 (1), 111–143.CrossRefGoogle Scholar
Egami, M. and Yamazaki, K. (2014) Phase-type fitting of scale functions for spectrally negative Lévy processes. Journal of Computational and Applied Mathematics, 264, 1–22.CrossRefGoogle Scholar
Hernández-Hernández, D., Pérez, J.L. and Yamazaki, K. (2016) Optimality of refraction strategies for spectrally negative Lévy processes. SIAM Journal on Control and Optimization, 54 (3), 1126–1156.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A.E. and Rivero, V. (2013) The theory of scale functions for spectrally negative Lévy processes. Lévy Matters II, Springer Lecture Notes in Mathematics, Berlin Heidelberg: Springer-Verlag.Google Scholar
Kyprianou, A.E. (2006) Fluctuations of Lévy Processes with Applications, 2nd edition. Berlin: Springer.Google Scholar
Kyprianou, A.E. and Loeffen, R. (2010) Refracted Lévy processes. Annales de l'Institut Henri Poincaré, 46 (1), 24–44.Google Scholar
Kyprianou, A.E., Loeffen, R. and Pérez, J.L. (2012) Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. Journal of Applied Probability, 49 (1), 150–166.CrossRefGoogle Scholar
Kyprianou, A.E., Pardo, J.C. and Pérez, J.L. (2014) Occupation times of refracted Lévy processes. Journal of Theoretical Probability, 27, 1292–1315.CrossRefGoogle Scholar
Loeffen, R. (2015) On obtaining simple identities for overshoots of spectrally negative Lévy processes. arXiv 1410.5341v2.Google Scholar
Pérez, J.L. and Yamazaki, K. (2015) On the refracted-reflected spectrally negative Lévy processes. arXiv 1511.06027.Google Scholar
Pérez, J.L. and Yamazaki, K. (2016) On the optimality of periodic barrier strategies for a spectrally positive Lévy process. arXiv 1604.07718.Google Scholar
Pistorius, M.R. (2003) On doubly reflected completely asymmetric Lévy processes. Stochastic Processes and their Applications, 1107 (1), 131–143.CrossRefGoogle Scholar
Pistorius, M.R. (2004) On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. Journal of Theoretical Probability, 17 (1), 183–220.CrossRefGoogle Scholar
Pistorius, M.R. (2007) An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. Seminaire de Probabilités XL, 287–307.CrossRefGoogle Scholar
Protter, P. (2005) Stochastic Integration and Differential Equations, 2nd edition. Berlin: Springer.CrossRefGoogle Scholar
Renaud, J.-F. (2014) On the time spent in the red by a refracted Lévy process. Journal of Applied Probability, 51, 1171–1188.CrossRefGoogle Scholar
Yin, C., Wen, Y. and Zhao, Y. (2013) On the optimal dividend problem for a spectrally positive Lévy process. Insurance: Mathematics and Economics, 53, 769–773.Google Scholar