  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.
