| It is well known that the optimal control of a stochastic system represents a general problem which can be found in many areas such as inventory control, financial engineering, and lately federal (national) reserve management, and so on. If the underlying system involves with some fixed transaction costs the problem will turn out to be known as an impulse control problem . It is well-known that the optimal solution to an impulse control problem can be sufficiently characterized by Quasi-Variational Inequality (QVI). With these profound findings and fundamental developments in impulse control theory, a mathematically rigid HJB-QVI system, which is formulated in the form of a functional boundary-value problem of Hamilton-Jacob-Bellman (HJB) equations, has been established as a general methodology for solving impulse control problems. In theory, optimal solution to a stochastic impulse control problem can be determined by solving a corresponding deterministic HJB-QVI system. However, in reality, HJB-QVI system of a practical impulse control problem is often too complicated to have an analytical solution in closed forms. As far as we can ascertain from the literature, apart from very few extremely simplified problems, closed-form analytical solution to an HJB-QVI system is seldom attainable. In this study, we obtain computational properties of the aforementioned QVI systems associated with impulse control problems, and provide computational methods for solving the QVI systems, which we categorize into two major classes: (1) QVI systems with analytically solvable HJB equations; (2) QVI systems with analytically unsolvable HJB equations. We begin with the study on the class-1 QVI systems. Although general solutions to underlying HJB equation of a class-1 QVI systems are obtainable, the associated QVI system may still need to be solved in non-closed forms. We present the solution for two class-1 type QVI systems in Chapter 2 and Chapter 3. For class-2 QVI systems, a computational optimization algorithm is presented in Chapter 4. In the last chapter we again consider a class-1 QVI system. It has a non-symmetric cost structure, which has particular application in mutual insurance reserve control problem. A novel computation algorithm is developed to determine numerically the optimal policy. |
