Calcilating optimal dividend payment, reinsurance, and investment strategies in a diffusion model

We consider the problem of the maximization of the total expected discounted amount of dividends paid by an insurance company up to the bankruptcy. It is assumed that the reinsurance is allowed and the wealth can be invested in a risky asset whose dynamics is described by the Black–Scholes model with random drift obeying the Ornstein–Uhlenbeck process. In accordance with the general scheme of dynamic programming, the problem is reduced to solving a Dirichlet problem for the corresponding Hamilton–Jacobi–Bellman equation in the half-plane. The problem is solved numerically by means of a monotone finite-difference scheme, which is proved to converge to a unique viscosity solution of the mentioned equation. We present and discuss the results of numerical experiments which indicate some nontrivial properties of optimal strategies.