  In this paper we study the problem of optimal dividend payment strategy which
maximizes the expected discounted sum of dividends to a multidimensional set up
of n associated insurance companies where the surplus process follows an
n-dimensional compound Poisson process. The general manager of the companies
has the possibility at any time to exercise an irreversible switch into another
regime; we also take into account an expected discounted value at ruin. This
multidimensional dividend problem is a mixed singular control/optimal problem.
We prove that the optimal value function is a viscosity solution of the
associated HJB equation and that it can be characterized as the smallest
viscosity supersolution. The main contribution of the paper is to provide a
numerical method to approximate (locally uniformly) the optimal value function
by an increasing sequence of sub-optimal value functions of admissible
strategies defined in an n-dimensional grid. As a numerical example, we present
the optimal time of merger for two insurance companies.
