  We consider a two-dimensional optimal dividend problem in the context of two
insurance companies with compound Poisson surplus processes, who collaborate by
paying each other's deficit when possible. We solve the stochastic control
problem of maximizing the weighted sum of expected discounted dividend payments
(among all admissible dividend strategies) until ruin of both companies, by
extending results of univariate optimal control theory. In the case that the
dividends paid by the two companies are equally weighted, the value function of
this problem compares favorably with the one of merging the two companies
completely. We identify this optimal value function as the smallest viscosity
supersolution of the respective Hamilton-Jacobi-Bellman equation and provide an
iterative approach to approximate it numerically. Curve strategies are
identified as the natural analogue of barrier strategies in this
two-dimensional context. A numerical example is given for which such a curve
strategy is indeed optimal among all admissible dividend strategies, and for
which this collaboration mechanism also outperforms the suitably weighted
optimal dividend strategies of the two stand-alone companies.
