5 - Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model

Abstract

In this chapter, we consider the classical risk model where an insurance company is able to adjust a franchise amount continuously. The problem of optimal control by the franchise amount is solved from the viewpoint of survival probability maximization. We derive the Hamilton–Jacobi–Bellman equation for the optimal survival probability and prove the existence of the solution to this equation with certain properties. The verification theorem gives the connection between this solution and the optimal survival probability, which differ in a constant multiplier. Then, we concentrate on the case of exponentially distributed claim sizes. Finally, we extend these results to the problem of optimal control by a deductible amount.