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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] S. Asmussen , F. Avram and M.R. Pistorius (2004) Russian and American put options under exponential phase-type Lévy models. Stochastic Processes and Their Applications, 109 (1), 79111.

[2] B. Avanzi and H.U. Gerber (2008) Optimal dividends in the dual model with diffusion. Astin Bulletin, 38 (2), 653667.

[3] B. Avanzi , H.U. Gerber and E.S.W. Shiu (2007) Optimal dividends in the dual model. Insurance Mathematics and Economics, 41 (1), 111123.

[5] F. Avram , Z. Palmowski and M.R. Pistorius (2007) On the optimal dividend problem for a spectrally negative Lévy process. Annals of Applied Probability, 17 (1), 156180.

[6] P. Azcue and N. Muler (2005) Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Mathematics Finance, 15 (2), 261308.

[7] E. Bayraktar and M. Egami (2008) Optimizing venture capital investments in a jump diffusion model. Mathematical Methods of Operations Research, 67 (1), 2142.

[8] E. Biffis and A.E. Kyprianou (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] T. Chan , A. Kyprianou and M. Savov (2011) Smoothness of scale functions for spectrally negative Lévy processes. Probability Theory and Related Fields, 150, 691708.

[12] A. Kuznetsov , A. Kyprianou and V. Rivero (2013) The theory of scale functions for spectrally negative levy processes. Springer Lecture Notes in Mathematics, 2061, 97186.

[14] A.E. Kyprianou and V. Rivero (2008) Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electronics Journal of Probability, 13 (57), 16721701.

[15] R.L. Loeffen (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] M.R. Pistorius (2003) On doubly reflected completely asymmetric Lévy processes. Stochastic Processes and Their Application, 107 (1), 131143.

[17] B.A. Surya (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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