The optimal control problem with terminal condition and random intervention times

The impulse optimal control problem is an important research area in recent years. Baccarin [1] discussed the optimal control of a multidimensional cash management system where the cash balances fluctuated as a homogeneous diffusion process in Rn. They formulated the model as an impulse control problem on an unbounded domain with unbounded cost functions. Under general assumptions they characterized the value function as a weak solution of a quasivariational inequality in a weighted Sobolev space and they showed the existence of an optimal policy. Meng and Siu [3] investigated an optimal reinsurance and dividend problem of an insurance company with the presence of reinvestments, or retained earnings, they considered the general situation that the company needed to pay both fixed and proportional costs as mixed classical and impulse control problems to get the value function and the optimal strategy. Yao, Yang and Wang [7] considered the dividend payments and capital injections control problem in a dual risk model. This leaded to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem was solved. Numerical solutions were provided to illustrate the idea and methodologies, and some interesting economic insights were included. The stochastic control problems with random intervention times were originally put forward by Rogers and Zane in 1998 [4]. They discussed a simple model of liquidity effects by the complexity of Log normal controlled state, there was no closed-form solution for corresponding cost problem, but they established certain qualitative features of the solution. Wang [5] simplified the controlled state, got explicit solutions about a quadratic deviation cost and a proportional control cost both discounted problem and the ergodic problem. Yao etc [7] introduced a diffusion in controlled state based on Wang’s, discussed optimal control policies and value functions. The main feature of control problems with random intervention times is that the control is discrete, and control can only be exerted at the arrival times of the Poisson process N . This paper generalized state process and cost function, under some assumptions, we obtained more general results.