Stochastic Optimal Control Models for the Insurance Company with Bankruptcy Return

Abstract

In this paper we consider the optimal control problem for a insurance company. Our objective is to maximize the expectation of discounted dividends and its terminal value which represents the company liquidation value upon the time of bankruptcy. The surplus of the insurance company is governed by the Brownian motion with a constant drift and a diffusion term. The company can manage its risk exposure simultaneously through proportional reinsurance. Apart from the proportional reinsurance, the insurance company also pays out dividends with bounded dividends rate. With the help of the stochastic dynamic programming approach, we solve the control problem of maximizing the expectation of discounted dividends and the terminal value. We first construct a solution to the HJB equation and then verify that the solution of the HJB equation is indeed the optimal value function for our problem. We also give explicit expressions of the optimal strategies.