  We consider the optimal dividend problem in the so-called degenerate
bivariate risk model under the assumption that the surplus of one branch may
become negative. More specific, we solve the stochastic control problem of
maximizing discounted dividends until simultaneous ruin of both branches of an
insurance company by showing that the optimal value function satisfies a
certain Hamilton-Jacobi-Bellman (HJB) equation. Further, we prove that the
optimal value function is the smallest viscosity solution of said HJB equation,
satisfying certain growth conditions. Under some additional assumptions, we
show that the optimal strategy lies within a certain subclass of all admissible
strategies and reduce the two-dimensional control problem to a one-dimensional
one. The results are illustrated by a numerical example and Monte-Carlo
simulated value functions.
