Studies On Dividend Payments And Some Related Stochastic Control Problems

In operations research, optimization was introduced a long time ago. A natural field of the application of control techniques is mathematical finance. Because economic prob-lems usually are optimization problems. The business of manager of company is to make decision, which should be taken in an optimal way. In the field of mathematical finance, the early works using the optimization techniques include Merton (1969), Merton(1971), Karatzas (1997), Karatzas and Shreve (1997) and so on. An optimal dividend control problem was initially formulated by de Finetti (1957) and solved by Gerber (1969). The corresponding problem for a diffusion approximation has been solved by Shreve, Lehoczky and Gaver (1984). Recently, optimization problem has become a more and more heated topic in the field of insurance and finance. And a mass of theories and methods have been achieved. However, Hipp (2004) and Schmidi (2008) are two important summary articles on this topic.In this thesis, the problems considered start with some stochastic process Xtπcon-trolled by a stochastic processπ. To each initial value x and each admissible control process{πt} we define a performance function V(x,π). We are interested in determining the maximal performance function V(x)= maxπV(x,π) called the value function. Two questions then arise:What is the value function V(x) and-if it exists at all-what is the optimal control process{πt}, i.e., the control process leading to the value function V(x)= V(x,π)? In ord to get the value function, one has to make several assumptions on it and obtain the Hamilton-Jacobi-Bellman (HJB) equation satisfied by the unknown value function. Then, we try to prove that a possible solution to the equation really is the value function. The whole thesis is divided into six chapters. The first chapter introduces some essential background and main results.In Chapter 2, we study optimal dividend control problems in a compound Pois-son risk process. We add to the classical objective function a term related to the deficit at ruin. It establishes that the new defined objective function can be characterized by the Hamilton-Jacobi-Bellman equations and band strategies with modified bands arc still optimal. Closed form solutions are derived for some special distribution functions. Some existing results in Section 2.4 of Schmidli (2008) are extended.In Chapter 3, we model the asset values Xt by a geometric Brownian motion pro-cess. We assume that the company must keep its asset value above the default threshold m(m> 0) to meet the regulation requirement. The manager of the company can control its asset by paying dividends and issuing new equity with proportional transaction costs in the financial market. Our objective is to maximize the expected present value of the dividends minus the equity issuance until the asset values Xt falls to m. We identify the value functions by some control techniques. Finally, the optimal control prolem subject with both fixed and proportional costs is considered, some numerial examples are pro-vided.In the framework of the dual model, similar dividend payment-capital injection con-trol problems are studied in Chapter 4 and Chapter 5. Both proportional and fixed trans-action costs are taken into account, which lead to the optimal stopping time problems and impulse control problems, respectively. Unlike Chapter 4, we study the optimization problem under the constrain that only dividend strategies with dividend rate bounded by a ceiling are admissible in Chapter 5. The optimal strategies under two kinds of situations are compared, numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.In Chapter 6, we make the assumption that a currency with exchange rate dynamics follows a geometric mean reverting (GMR) process. The objective of the Central Bank is to keep this exchange rate as close as possible to a given target. The running cost associated to the difference between the exchange rate and the target is considered. Ad-ditionally, there are also fixed and proportional costs associated with each intervention. We aim at finding the optimal intervention strategy to minimize the total cost. This mixed classical-impulse stochastic control problem is studied.