Optimal Stochastic Control Policy of Discounted Problems with Quadratic Cost in Investment

The optimal singular stochastic control model plays an important role in making decisions for managers. Traditional model only considers the cost pushing towards the target with the same expense ratio, which differs from the reality. This paper proposes a new form of the expected system costs on discounted problems about a firm that wants to keep a target state, in which different expense ratios are assigned to the two kinds of cost related to investment and disinvestment behavior. The question is how to minimize the total expected cost of both ´action´ and ´deviation from a target state 0´. The answer to the question takes the form of exerting control in a singular manner, in order not to exit from certain region. In order to obtain the minimum of this cost, by using optimal stochastic control theory, we get the optimal policy and its corresponding total cost value which is explicitly shown in an analytical expression. Furthermore, we present some numerical examples for some concrete parameters.