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Optimal dividend strategies for two collaborating insurance companies

Part of: Stochastic systems and control Mathematical economics Hamilton-Jacobi theories, including dynamic programming

Published online by Cambridge University Press:  26 June 2017

Hansjörg Albrecher ,
Pablo Azcue  and
Nora Muler
Hansjörg Albrecher*
Affiliation:
University of Lausanne and Swiss Finance Institute
Pablo Azcue*
Affiliation:
Universidad Torcuato Di Tella
Nora Muler*
Affiliation:
Universidad Torcuato Di Tella
*
* Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, Switzerland. Email address: hansjoerg.albrecher@unil.ch
** Postal address: Departamento de Matematicas, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, C1428BCW Buenos Aires, Argentina.
** Postal address: Departamento de Matematicas, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, C1428BCW Buenos Aires, Argentina.
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Abstract

We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

Keywords

Compound Poisson processtwo-dimensional optimal dividend problemHamilton–Jacobi–Bellman equationviscosity solutions

MSC classification

Primary: 91B30: Risk theory, insurance 93E20: Optimal stochastic control 49L25: Viscosity solutions
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 515 - 548
Copyright
Copyright © Applied Probability Trust 2017 

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