An optimal investment strategy for insurers in incomplete markets

In this paper we consider the problem of an insurance company where the wealth of the insurer is described by a Cramér-Lundberg process. The insurer is allowed to invest in a risky asset with stochastic volatility subject to the influence of an economic factor and the remaining surplus in a bank account. The price of the risky asset and the economic factor are modeled by a system of correlated stochastic differential equations. In a finite horizon framework and assuming that the market is incomplete, we study the problem of maximizing the expected utility of terminal wealth. When the insurer's preferences are exponential, an existence and uniqueness theorem is proven for the non-linear Hamilton-Jacobi-Bellman equation (HJB). The optimal strategy and the value function have been produced in closed form. In addition and in order to show the connection between the insurer's decision and the correlation coefficient we present two numerical approaches: A Monte-Carlo method based on the stochastic representation of the solution of the insurer problem via Feynman-Kac's formula, and a mixed Finite Difference Monte-Carlo one. Finally the results are presented in the case of Scott model.