Optimal Reinsurance under Terminal Value Constraints
Optimal reinsurance is a perennial problem in insurance. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some important distinctions. In particular, the surplus of an insurance company is routinely approximated by a Brownian motion, as opposed to the geometric Brownian motion to model assets in finance. Furthermore, exposure to risk is controlled via reinsurance, rather than risky investments. This leads to interesting qualitative differences in the optimal solutions. In this paper, using the martingale method, we derive the optimal proportional, non cheap reinsurance control that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider a number of realistic constraints on the terminal value: the strict lower boundary, the probability (Value at Risk) constraint, and the expected shortfall (conditional Value at Risk) constraints under the P and Q measures, respectively.