Optimization for stochastic model arisen from investment problem associated with default risk

In this paper, we study the optimal strategy arisen from the excess-of-loss reinsurance and asset allocation in defaultable markets under a general stochastic model. By developing the corresponding dynamic programming approach, we establish the optimal investment approach through two sub-problems: a pre-default case and a post-default case, respectively, characterized by the obtained Hamilton–Jacobi–Bellman (HJB) equations. We show the existence of a classical solution to the pre-default case via super-sub solution techniques and give an explicit characterization of the optimal reinsurance and investment policy that maximizes the common used utility associated with the terminal wealth. Verification theorem is further established to show the uniqueness of the corresponding solution of HJB equation that is critical to the desired optimal solution.