A mathematical model of the optimal premium policy of an insurance firm with delay

In this work, we study the optimization problem confronted by an insurance firm whose management can control its cash-balance dynamics by adjusting the under- lying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels by selecting an appropriate pre- mium policy. We study the problem in a general framework and make two interesting inclusions; Firstly, we introduce the aspect of time delay to the system; and we replace the stan- dard expected additive utility function with a Stochastic differential utility (SDU). We reformulate the problem and obtain a semi coupled Forward-Backward Stochas- tic Differential Equation and prove that this forward-backward system admits a unique solution. The Sufficient Maximum Principle for an optimal control of such a system is established and we conclude with a case study of three particular cases which fit into our general model. Some concluding remarks are also given.