Optimal dividend strategies for a catastrophe insurer
We assume that the free surplus of the insurance company follows a compound Cox process generated by a shot-noise intensity, modelling the arrival of claims due to catastrophic events. The goal is to find the dividend payment strategy that maximizes the expected discounted dividends until ruin. This optimal control problem is two-dimensional because it depends on both the current surplus and the current intensity of the arrivals of claims. We characterize the optimal value function as the smallest viscosity solution of the corresponding two-dimensional Hamilton-Jacobi-Bellman equation. It is shown that the optimal value function can be uniformly approximated through a discretization of the space of the free surplus and current claim intensity level. We implement the resulting numerical scheme to identify optimal dividend strategies, 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 regions (with lump sum dividend payments) and non-action regions (no dividend payments) as a function of the current surplus and intensity level. We also discuss some interpretations and investigate the upward potential for shareholders when including a catastrophe sector in the portfolio. This is joint work with Hansjoerg Albrecher.