Asymptotic Behavior of an Optimal Investment-Reinsurance Problem with General Utility Functions

Abstract

It is usually extremely difficult to derive an analytical solution to the Hamilton-Jacobi-Bellman (HJB) equation for some optimal control problems under general utility functions, while this paper provides a dual control method to solve the HJB equation for an optimal investment-reinsurance problem with general utility functions. We first change the HJB equation into its dual one. By solving the solution to the dual HJB equation, we derive a classical solution to the primal HJB equation and obtain the expression of the optimal investment-reinsurance strategy, then some examples, which would be difficult to solve with the standard method, are given to demonstrate the usefulness of our dual control method. Moreover, applying the dual method along with the partial differential equation method, we verify the asymptotic behavior of the optimal control strategy, which means that the optimal investment-reinsurance strategy with the general utility functions converges to that with the power utility function at any level of the initial wealth.