Optimal dividend payments for a two-dimensional insurance risk process

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 premiums 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 branch. 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 analyze some numerical examples.