Optimal proportional reinsurance and investment for stochastic factor models

In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramér–Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. The financial market is supposed not influenced by the stochastic factor, hence it is independent on the insurance market. Using the classical stochastic control approach based on the Hamilton–Jacobi–Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions to two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations are performed to obtain sensitivity analyses.