Optimal Dividend Strategies for Two Collaborating Insurance Companies

TL;DR

This paper develops a two-dimensional stochastic control model for two collaborating insurance companies to optimize dividend payouts, demonstrating that collaborative strategies can outperform independent or merged approaches.

Contribution

It extends univariate optimal dividend control theory to a two-company setting with collaboration, identifying optimal strategies and providing numerical approximation methods.

Findings

Collaborative dividend strategies outperform independent strategies.

Curve strategies serve as optimal analogues to barrier strategies.

Numerical examples confirm the effectiveness of collaboration mechanisms.

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 solve 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 this 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.