Optimal investment and proportional reinsurance for a jump–diffusion risk model with constrained control variables

This paper considers the optimal control problem with constraints for an insurer. The risk process is assumed to be a jump–diffusion process, and the risk can be reduced through a proportional reinsurance. In addition, the surplus can be invested in the financial market consists of one risk-free asset and one risky asset. The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term’s explanations. In all cases, with normal constraints on the control variables, the value functions and the corresponding optimal strategies are given in a closed form. Numerical simulations are presented to illustrate the effects of parameters on the optimal strategies as well as the economic meaning behind.