Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T16:15:04.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimality of refraction strategies for a constrained dividend problem

Part of: Mathematical economics Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  03 September 2019

Mauricio Junca ,
Harold A. Moreno-Franco ,
José Luis Pérez  and
Kazutoshi Yamazaki
Mauricio Junca*
Affiliation:
Universidad de los Andes
Harold A. Moreno-Franco*
Affiliation:
Universidad del Norte and HSE University
José Luis Pérez*
Affiliation:
Centro de Investigación en Matemáticas
Kazutoshi Yamazaki*
Affiliation:
Kansai University
*
*Postal address: Department of Mathematics, Universidad de los Andes, Carrera 1 No. 18A–12, CP 11711, Bogotá, Colombia. Email address: [email protected]
**Postal address: Department of Mathematics and Statistics, Universidad del Norte, Km. 5 Vía Puerto Colombia, CP 080003, Barranquilla, Colombia. Email address: [email protected]
***Postal address: Department of Probability and Statistics, Centro de Investigación en Matemáticas A. C. Calle Jalisco s/n., CP 36240, Guanajuato, Mexico. Email address: [email protected]
****Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamatecho, Suita-shi, Osaka 564-8680, Japan. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

Keywords

Dividend paymentoptimal controlruin time constraintspectrally one-sided Lévy processrefracted Lévy processscale function

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 93E20: Optimal stochastic control 91B30: Risk theory, insurance
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 3 , September 2019 , pp. 633 - 666
Copyright
© Applied Probability Trust 2019 

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

Avanzi, B., Pérez, J.-L., Wong, B. and Yamazaki, K. (2017). On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models. Insurance Math. Economics 72, 148–162.CrossRefGoogle Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.CrossRefGoogle Scholar
Azcue, P. and Muler, N. (2012). Optimal dividend policies for compound Poisson processes: the case of bounded dividend rates. Insurance Math. Economics 51, 26–42.CrossRefGoogle Scholar
Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359–372.CrossRefGoogle Scholar
Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22.CrossRefGoogle Scholar
Grandits, P. (2015). An optimal consumption problem in finite time with a constraint on the ruin probability. Finance Stoch. 19, 791–847.CrossRefGoogle Scholar
Hernández, C. and Junca, M. (2015). Optimal dividend payments under a time of ruin constraint: exponential claims. Insurance Math. Economics 65, 136–142.CrossRefGoogle Scholar
Hernández, C., Junca, M. and Moreno-Franco, H. (2018). A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes. Insurance Math. Economics 79, 57–68.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.CrossRefGoogle Scholar
Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 24–44.CrossRefGoogle Scholar
Kyprianou, A. E., Loeffen, R. and Pérez, J.-L. (2012). Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Prob. 49, 150–166.CrossRefGoogle Scholar
Kyprianou, A. E., Rivero, V. and Song, R. (2010). Convexity and smoothness of scale functions and de Finetti’s control problem. J. Theoret. Prob. 23, 547–564.CrossRefGoogle Scholar
Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669–1680.CrossRefGoogle Scholar
Loeffen, R. L. (2009). An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. J. Appl. Prob. 46, 85–98.CrossRefGoogle Scholar
Noba, K., Pérez, J.-L., Yamazaki, K. and Yano, K. (2018). On optimal periodic dividend strategies for Lévy risk processes. Insurance Math. Economics 80, 29–44.CrossRefGoogle Scholar
Paulsen, J. (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance Stoch. 7, 457–473.CrossRefGoogle Scholar
Pérez, J.-L., Yamazaki, K. and Yu, X. (2018). On the bail-out optimal dividend problem. J. Optimization Theory Appl. 179, 553–568.CrossRefGoogle Scholar
Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.CrossRefGoogle Scholar
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Prob. 12, 890–907.Google Scholar
Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Economics 41, 163–184.CrossRefGoogle Scholar
Yin, C., Wen, Y. and Zhao, Y. (2014). On the optimal dividend problem for a spectrally positive Lévy processes. ASTIN Bull. 44, 635–651.CrossRefGoogle Scholar