Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

This paper studies the optimal proportional reinsurance and investment strategy for an insurer who invests one paired assets, where their price spread is described by Ornstein-Uhlenbeck (O-U) processes. The insurer's objective is to maximize the expected exponential utility of the terminal wealth in a finite time horizon under two risk models: a classical risk model and a diffusion model. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we characterize the optimal strategies and provide a verification result for the value function via the exponential integrability of the square of an O-U process. Finally, numerical examples are performed to obtain sensitivity analysis.