Author:  Yuan, Jiguang 
Title:  Computational optimization of mutual insurance systems: a QuasiVariational Inequality approach 
Degree:  Ph.D. 
Year:  2009 
Subject:  Hong Kong Polytechnic University  Dissertations. Insurance. Variational inequalities (Mathematics) 
Department:  Dept. of Logistics and Maritime Studies 
Pages:  ix, 128 leaves : ill. ; 30 cm. 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2321635 
URI:  http://theses.lib.polyu.edu.hk/handle/200/3643 
Abstract:  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. The framework of solving this type of problem is initially developed by Bensoussan [10] and Aubin [5]. They proved that the optimal solution to an impulse control problem can be sufficiently characterized by QuasiVariational Inequality (QVI). With these profound findings and fundamental developments in impulse control theory, a mathematically rigid HJBQVI system (deterministic), which is formulated in the form of a functional boundaryvalue problem of nonlinear HamiltonJacobBellman (HJB) equations, has been established as a general methodology for solving impulse control problems (stochastic). In theory, optimal solution to a stochastic impulse control problem can be determined by solving a corresponding deterministic HJBQVI system. However, in reality, HJBQVI 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 [34, 11, 28], closedform analytical solution to an HJBQVI 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 class1 QVI systems. Although general solutions to underlying HJB equation of a class1 QVI systems are obtainable, the associated QVI system may still need to be solved in nonclosed forms. We present the solution for two class1 type QVI systems in Chapter 2 and Chapter 3. For class2 QVI systems, to obtain numerical solutions a computational optimization algorithm presented in Chapter 4. In the last chapter we again consider a class1 QVI system. It has a nonsymmetric cost structure, which has particular application in mutual insurance reserve control problem. A novel computation algorithm is developed to determine numerically an optimal (a, A; B, b) policy. 
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