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  • Erhan Bayraktar (a1), Andreas E. Kyprianou (a2) and Kazutoshi Yamazaki (a3)

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.

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[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), 79111.
[2]Avanzi, B. and Gerber, H.U. (2008) Optimal dividends in the dual model with diffusion. Astin Bulletin, 38 (2), 653667.
[3]Avanzi, B., Gerber, H.U. and Shiu, E.S.W. (2007) Optimal dividends in the dual model. Insurance Mathematics and Economics, 41 (1), 111123.
[4]Avanzi, B., Shen, J. and Wong, B. (2011) Optimal dividends and capital injections in the dual model with diffusion. Astin Bulletin, 41 (2), 611644.
[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), 156180.
[6]Azcue, P. and Muler, N. (2005) Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Mathematics Finance, 15 (2), 261308.
[7]Bayraktar, E. and Egami, M. (2008) Optimizing venture capital investments in a jump diffusion model. Mathematical Methods of Operations Research, 67 (1), 2142.
[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), 8591.
[9]Chan, T., Kyprianou, A. and Savov, M. (2011) Smoothness of scale functions for spectrally negative Lévy processes. Probability Theory and Related Fields, 150, 691708.
[10]Egami, M. and Yamazaki, K. (2012) Phase-type fitting of scale functions for spectrally negative Lévy processes. arXiv:1005.0064.
[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. 119146. Progress in Probability, Birkhäuser.
[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, 97186.
[13]Kyprianou, A.E. (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer-Verlag.
[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), 16721701.
[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), 16691680.
[16]Pistorius, M.R. (2003) On doubly reflected completely asymmetric Lévy processes. Stochastic Processes and Their Application, 107 (1), 131143.
[17]Surya, B.A. (2008) Evaluating scale functions of spectrally negative Lévy processes. Journal of Applied Probability, 45 (1), 135149.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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