A two-dimensional dividend problem for collaborating companies and an optimal stopping problem

Peter Grandits

doi:10.1080/03461238.2018.1498387

Abstract

ABSTRACTWe consider two insurance companies with wealth processes described by two independent Brownian motions with drift. The goal of the companies is to maximize their expected aggregated discounted dividend payments until ruin. The companies are allowed to help each other by means of transfer payments. But in contrast to Gu et al. [(2018). Optimal dividend strategies of two collaborating businesses in the diffusion approximation model. Mathematics of Operations Research 43(2), 377–398], they are not obliged to do so, if one company faces ruin. We show that the problem is equivalent to a mixture of a one-dimensional singular control problem and an optimal stopping problem. The value function is explicitly constructed and a verification result is proved. Moreover, the optimal strategy is provided as well.

Highlights

Since B. de Finnetti has suggested in his seminal paper (de Finetti 1957) an alternative to the classical ruin probability approach to evaluate an insurance company, namely the maximization of expected discounted dividends paid by the company, there has been a lot of research in this direction
We consider two insurance companies with wealth processes described by two independent Brownian motions with drift
We show that the problem is equivalent to a mixture of a one-dimensional singular control problem and an optimal stopping problem

Summary

Introduction

Since B. de Finnetti has suggested in his seminal paper (de Finetti 1957) an alternative to the classical ruin probability approach to evaluate an insurance company, namely the maximization of expected discounted dividends paid by the company, there has been a lot of research in this direction. Gerber proved in Gerber (1969) that, if the underlying wealth process follows the classical CramerLundberg model, it is optimal to use a so called band strategy, which degenerates in the case of exponential individual claims to a barrier strategy, meaning that you have to keep your endowment just below a certain barrier value by paying dividends (see Schmidli 2008 for a different proof of these facts). Our paper can be considered as a singular stochastic control problem, subject to discretionary stopping. Some articles in this direction are Davis and Zervos (1994), Davis and Zariphopoulou (1995), Karatzas and Wang (2000), Karatzas et al (2000), Karatzas and Zamfirescu (2006) and Zervos (2003).

The model and preliminary results

Explicit construction of the value function

The verification result and the optimal strategy