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Optimality of refraction strategies for a constrained dividend problem

Part of: Stochastic processes Mathematical economics 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: mj.junca20@uniandes.edu.co
**Postal address: Department of Mathematics and Statistics, Universidad del Norte, Km. 5 Vía Puerto Colombia, CP 080003, Barranquilla, Colombia. Email address: hamoreno@uninorte.edu.co
***Postal address: Department of Probability and Statistics, Centro de Investigación en Matemáticas A. C. Calle Jalisco s/n., CP 36240, Guanajuato, Mexico. Email address: jluis.garmendia@cimat.mx
****Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamatecho, Suita-shi, Osaka 564-8680, Japan. Email address: kyamazak@kansai-u.ac.jp
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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 

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