  In this paper we study the problem of optimally paying out dividends from an
insurance portfolio, when the criterion is to maximize the expected discounted
dividends over the lifetime of the company and the portfolio contains claims
due to natural catastrophes, modelled by a shot-noise Cox claim number process.
The optimal value function of the resulting two-dimensional stochastic control
problem is shown to be the smallest viscosity supersolution of a corresponding
Hamilton-Jacobi-Bellman equation, and we prove that it can be uniformly
approximated through a discretization of the space of the free surplus of the
portfolio and the current claim intensity level. We implement the resulting
numerical scheme to identify optimal dividend strategies for such a natural
catastrophe insurer, and it is shown that the nature of the barrier and band
strategies known from the classical models with constant Poisson claim
intensity carry over in a certain way to this more general situation, leading
to action and non-action regions for the dividend payments as a function of the
current surplus and intensity level. We also discuss some interpretations in
terms of upward potential for shareholders when including a catastrophe sector
in the portfolio.
