Mean-avoiding Ornstein-Uhlenbeck process and unrestricted dividends

We consider an insurance company endowed with an initial capital and a surplus driven by a mean-avoiding Ornstein-Uhlenbeck (OU) process. The company’s target is to maximise the expected discounted unrestricted dividends. The parameters of the considered OU process are crucial for finding the optimal strategy. To solve the problem, we distinguish between three different cases. In the first case, the absolute value of the speed rate of the OU process is assumed to be strictly smaller than the constant discounting rate. Here, we are able to find both the optimal strategy – a bang-bang strategy with a constant barrier – and the value function. If the absolute value of the speed is equal to the discounting rate, we find an explicit expression for the value function and prove that the optimal strategy does not exist. In the last case, when the absolute value of the speed rate exceeds the discounting rate, the value function is infinite.