REFRACTION–REFLECTION STRATEGIES IN THE DUAL MODEL

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.

[1]  Kazutoshi Yamazaki,et al.  Phase-type fitting of scale functions for spectrally negative Lévy processes , 2010, J. Comput. Appl. Math..

[2]  Mladen Savov,et al.  Smoothness of scale functions for spectrally negative Lévy processes , 2009, 0903.1467.

[3]  Andreas E. Kyprianou,et al.  The Theory of Scale Functions for Spectrally Negative Lévy Processes , 2011, 1104.1280.

[4]  Anja Sturm,et al.  Stochastic Integration and Differential Equations. Second Edition. , 2005 .

[5]  Andreas E. Kyprianou,et al.  Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes , 2010, Journal of Applied Probability.

[6]  Daniel Hernández-Hernández,et al.  Optimality of Refraction Strategies for Spectrally Negative Lévy Processes , 2016, SIAM J. Control. Optim..

[7]  C. Yin,et al.  Optimal dividends problem with a terminal value for spectrally positive Levy processes , 2013, 1302.6011.

[8]  Jean-François Renaud.  On the Time Spent in the Red by a Refracted Lévy Risk Process , 2014, J. Appl. Probab..

[9]  M. Pistorius.  On doubly reflected completely asymmetric Lévy processes , 2003 .

[10]  Bernard Wong,et al.  On Optimal Periodic Dividend Strategies in the Dual Model with Diffusion , 2013 .

[11]  J. C. Pardo,et al.  Occupation Times of Refracted Lévy Processes , 2012, 1205.0756.

[12]  On obtaining simple identities for overshoots of spectrally negative L\'evy processes , 2014, 1410.5341.

[13]  M. Pistorius,et al.  On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum , 2004 .

[14]  C. Yin,et al.  ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS , 2013, ASTIN Bulletin.

[15]  Erhan Bayraktar,et al.  Optimal Dividends in the Dual Model Under Transaction Costs , 2013, 1301.7525.

[16]  Benjamin Avanzi,et al.  Optimal dividends in the dual model , 2007 .

[17]  José-Luis Pérez,et al.  On the Refracted-Reflected Spectrally Negative Levy Processes , 2015 .

[18]  A. Kyprianou.  Fluctuations of Lévy Processes with Applications , 2014 .

[19]  Martijn Pistorius,et al.  An Excursion-Theoretical Approach to Some Boundary Crossing Problems and the Skorokhod Embedding for Reflected Lévy Processes , 2007 .

[20]  Andreas E. Kyprianou,et al.  Refracted Lévy processes , 2008 .

[21]  Florin Avram,et al.  On the optimal dividend problem for a spectrally negative Lévy process , 2007, math/0702893.

[22]  Kazutoshi Yamazaki,et al.  Precautionary measures for credit risk management in jump models , 2010, 1004.0595.

[23]  Refracted Lévy processes , 2008 .

[24]  Kazutoshi Yamazaki,et al.  On the optimality of Periodic barrier strategies for a spectrally positive L\'evy process , 2016 .

[25]  P. Protter.  Stochastic integration and differential equations , 1990 .

[26]  S. Asmussen,et al.  Russian and American put options under exponential phase-type Lévy models , 2004 .

[27]  Benjamin Avanzi,et al.  Optimal Dividends and Capital Injections in the Dual Model with Diffusion , 2010 .

[28]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[29]  Benjamin Avanzi,et al.  Optimal Dividends in the Dual Model with Diffusion , 2008 .