Optimal dividend payments for a two-dimensional insurance risk process

TL;DR

This paper addresses the optimal dividend distribution problem for a two-branch insurance company modeled by a two-dimensional compound Poisson process, providing a solution framework and analyzing numerical examples.

Contribution

It introduces a novel two-dimensional stochastic control model for insurance dividends and characterizes the optimal strategy using viscosity solutions.

Findings

The value function is the smallest viscosity supersolution of the HJB equation.

Optimal dividend strategies are explicitly described.

Numerical examples illustrate the theoretical results.

Abstract

We consider a two-dimensional optimal dividend problem in the context of two branches of an insurance company with compound Poisson surplus processes dividing claims and premia in some specified proportions. We solve the stochastic control problem of maximizing expected cumulative discounted dividend payments (among all admissible dividend strategies) until ruin of at least one company. We prove that the value function is the smallest viscosity supersolution of the respective Hamilton-Jacobi-Bellman equation and we describe the optimal strategy. We analize some numerical examples.