Studies on Dynamic Optimization of Stochastic Systems with Multiple Criteria

首先，我研究了一个具有泊松跳跃的几何布朗运动的最优停止问题。尽管有人认为所谓的平滑粘贴技术(参见Dixit（1993)，以及Dixit和Pindyck(1994)）对这种随机优化问题很有用，但迄今为止，似乎还没有充分讨论其数学有效性。在这个项目中，通过采用鞅方法，我证明了在一组温和的条件下，在问题的参数上，它确实是数学上有效的。随机优势（随机顺序）和不等式在经济和金融的各个领域是非常有用的工具。这个项目的第二个目的是描述迄今为止通过在财务优化中使用随机优势的思想获得的主要结果。重点研究了投资组合选择中的需求效应和转移效应问题。本文还收集了一些与优化问题没有直接关系的其他例子，以展示随机优势在金融中的广泛应用领域。此外，由于在可靠性和可维护性理论中已知的几个随机优势和相关不等式，甚至在金融理论中也非常有用，本项目的下一个目的是简要概述有关随机顺序及其在可靠性和可维护性理论的各个领域中发展的应用的有用的已知结果。

First, I examined an optimal stopping problem for a geometric Brownian motion with Poissonian jumps. Although it has been argued that so called smooth pasting technique (see Dixit (1993), and Dixit and Pindyck (1994)) is useful for such stochastic optimization problems, it seems that its mathematical validity is not sufficiently discussed so far. In this project, by taking a martingale approach, I showed that it is indeed mathematically valid under a set of some mild conditions on the parameters of the problem.Stochastic dominances (stochastic orders) and inequalities are very useful tools in various areas of economics and finance. The second purpose of this project was to describe main results obtained so far by using the idea of stochastic dominances in financial optimization. Especially, the emphasis is placed on the demand and shift effect problems in portfolio selection. Some other examples, which are not related directly to optimization problems, are also gathered to demonstrate the wide spectrum of application areas of stochastic dominances in finance. Further, since several stochastic dominances and related inequalities which are known in the reliability and maintainability theory, are very useful even in finance theory, the next purpose of this project was to provide a brief survey of the useful known results concerning stochastic orders and their applications developed in various areas of the reliability and maintainability theory.