Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T05:20:08.075Z Has data issue: false hasContentIssue false
  • English
  • Français

ON OPTIMAL DIVIDENDS IN THE DUAL MODEL

Published online by Cambridge University Press:  10 July 2013

Erhan Bayraktar ,
Andreas E. Kyprianou  and
Kazutoshi Yamazaki
Erhan Bayraktar*
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA
Andreas E. Kyprianou
Affiliation:
Department of Mathematical Sciences, The University of Bath, Claverton Down, Bath BA2 7AY, UK E-mail: a.kyprianou@bath.ac.uk
Kazutoshi Yamazaki
Affiliation:
Department of Mathematics, Faculty of Engineering Science, Kansai University, Suita-shi, Osaka 564-8680, Japan E-mail: kyamazak@kansai-u.ac.jp
*
E-mail: erhan@umich.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We revisit the dividend payment problem in the dual model of Avanzi et al. ([2–4]). Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.

Keywords

Dual modeldividendscapital injectionsspectrally positive Lévy processesscale functions
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 43 , Issue 3 , September 2013 , pp. 359 - 372
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © ASTIN Bulletin 2013

References

[1] 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.Google Scholar
[2] Avanzi, B. and Gerber, H.U. (2008) Optimal dividends in the dual model with diffusion. Astin Bulletin, 38 (2), 653–667.Google Scholar
[3] Avanzi, B., Gerber, H.U. and Shiu, E.S.W. (2007) Optimal dividends in the dual model. Insurance Mathematics and Economics, 41 (1), 111–123.Google Scholar
[4] 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
[5] 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 (1), 156–180.Google Scholar
[6] Azcue, P. and Muler, N. (2005) Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Mathematics Finance, 15 (2), 261–308.Google Scholar
[7] Bayraktar, E. and Egami, M. (2008) Optimizing venture capital investments in a jump diffusion model. Mathematical Methods of Operations Research, 67 (1), 21–42.Google Scholar
[8] Biffis, E. and Kyprianou, A.E. (2010) A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Mathematics and Economics, 46 (1), 85–91.Google Scholar
[9] Chan, T., Kyprianou, A. and Savov, M. (2011) Smoothness of scale functions for spectrally negative Lévy processes. Probability Theory and Related Fields, 150, 691–708.Google Scholar
[10] Egami, M. and Yamazaki, K. (2012) Phase-type fitting of scale functions for spectrally negative Lévy processes. arXiv:1005.0064.Google Scholar
[11] Hubalek, F. and Kyprianou, A.E. (2010) Old and new examples of scale functions for spectrally negative Lévy processes. In Sixth Seminar on Stochastic Analysis, Random Fields and Applications (eds. Dalang, R., Dozzi, M. and Russo, F.), pp. 119–146. Progress in Probability, Birkhäuser.Google Scholar
[12] Kuznetsov, A., Kyprianou, A. and Rivero, V. (2013) The theory of scale functions for spectrally negative levy processes. Springer Lecture Notes in Mathematics, 2061, 97–186.Google Scholar
[13] Kyprianou, A.E. (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer-Verlag.Google Scholar
[14] Kyprianou, A.E. and Rivero, V. (2008) Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electronics Journal of Probability, 13 (57), 1672–1701.Google Scholar
[15] Loeffen, R.L. (2008) On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Annals of Applied Probability, 18 (5), 1669–1680.Google Scholar
[16] Pistorius, M.R. (2003) On doubly reflected completely asymmetric Lévy processes. Stochastic Processes and Their Application, 107 (1), 131–143.Google Scholar
[17] Surya, B.A. (2008) Evaluating scale functions of spectrally negative Lévy processes. Journal of Applied Probability, 45 (1), 135–149.Google Scholar