Optimal premium policy of an insurance firm with delay and stochastic interest rate

In this paper, we study the optimization problem confronted by an insurance firm whose management can control its cash-balance dynamics by adjusting the underlying 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 premium policy. We study the problem in a general framework assuming the state process is governed by a stochastic delay differential equation and the classical utility function being replaced by a recursive utility or stochastic differential utility (SDU). We derive a sufficient maximum principle for an optimal control of such a system and apply the result to discuss some optimal premium rate control problems.