Optimal Controls for a Large Insurance Under a CEV Model: Based on the Legendre Transform Dual Method
The purpose of this paper is to consider the optimal proportional reinsurance and investment strategies for an insurance company. The insurer's surplus process is approximated by a Brownian motion with drift. The insurance company can purchase proportional reinsurance and invest the surplus in a financial market which includes one risk free asset and one risky asset whose price is modeled by a CEV model. The primary problem is changed to the dual problem by implying Legendre transform. When the objective of the insurance company is to maximize the expected logarithmic utility from terminal wealth, the closed form expressions for the optimal reinsurance investment policy which is different to the Merton case to the primal optimal problem are obtained and numerical simulations are provided to demonstrate our results. Moreover, we find an interesting result that risk exposure is non monotonic in the cost of reinsurance.