Optimal investment and reinsurance strategy

An insurance company is modelled by a compound Poisson process and it is assumed that the company has a possibility to purchase an excess of loss reinsurance defined by retention level as well as invest its surplus into a risky asset described by the Black–Scholes model. An optimal survival probability is derived as a solution to the corresponding Hamilton–Jacobi–Bellman equation. It is proved that any increasing solution to the Hamilton–Jacobi–Bellman equation defines the optimal strategy.