A Class Of Singular Stochastic Control Problems With Stopping Time

According to the state process, the stochastic control problem can be classified as following, the singular stochastic control and the impulse stochastic control; while according to the cost function, it contains the discounted cost problem, the stationary problem and the finite horizon problem. So far, the discounted cost problem of the singular stochastic control and the stationary problem of the impulse stochastic control have been solved perfectly. The stationary problem of the singular stochastic control and the discounted cost problem of the impulse stochastic control have been widely studied in general conditions. With the development of the optimal stochastic control problem, a class of singular stochastic control with stopping time has been studied, which contains a stopping time in the cost function. The problem is applied in many fields, especially in financial field, so it arose many attention. This paper tries to re-search the problem of singular stochastic control with stopping time. Both the state process and the cost function are generalized. In the general conditions, it gets the op-timal strategy, and it will be applied in more fields. Moreover, using the result we got, we solve a problem of investment strategy.This paper is organized as follows. Chapter 1 gives the introduction on the back-ground and development of the stochastic control problems. In chapter 2, we introduce some variational inequalities in the discounted cost problem of the singular stochastic control and the stationary problem of the impulse stochastic control. We also show that the solution of the variational inequalities and the cost function satisfy an inequality, which is important for getting the optimal strategy.From chapter 3, we start to research the discounted cost problem of the singular stochastic control with stopping time. In chapter 3, we introduce a drift parameter into the state and extend the diffusion parameter from 1 to?. We get the solution of the variational inequalities in two cases and give the optimal strategies. We also show the solution is just the optimal cost function. The model in chapter 4 is the extension of that in chapter 3. In chapter 4, the state process is the solution of a stochastic differential equation, and we also generalize the cost function. By solving the variational inequal-ities, we show the existence of the optimal control and optimal stopping time and give the exact expression of the optimal cost function. In chapter 5, we consider an invest- ment model which bases on a class of singular stochastic control with stopping time. The state process which means the commodity price is the add of a geometry Brown motion and a increasing process. According to the fluctuations of the commodity price, the investor could choose the optimal investment and exit time decision to minimize the cost. Using the result we have got, We show the existence of the optimal investment strategy and give the exact expression of the optimal cost function.Chapter 6 introduces the work we are carrying on. Consider an investment model that the investor could choose the expansion and contraction decision and choose the entry and exit time according to market demand fluctuations. The subject of investor is minimizing the cost of production. Moreover, we will study the impulse stochastic control problem with stopping time.