【作者】 息悦；

【导师】 罗成新；

【作者基本信息】 沈阳师范大学 ， 应用数学， 2011， 硕士

【摘要】 随着概率论和随机过程等近代数学理论的发展和应用,利用随机分析方法研究最优投资与消费问题已成为金融数学中定量研究的热门领域之一.Merton提出的连续时间下投资组合及消费模型标志着连续时间投资组合理论的真正开始.他通过把随机控制理论应用到最优投资组合问题中,得到了一些特殊情形下的显式解.本文研究证券市场中不同模型的随机最优投资组合问题.分别选取了常数相对风险厌恶函数、绝对风险厌恶函数和幂效用函数,通过运用直接构造的方法和动态规划的方法得到了最优投资策略,并给出了相应的值函数.本文的具体安排如下.第一章介绍最优投资组合问题的背景、发展状况以及研究意义.比较系统地给出了关于最优投资组合问题的一些基础知识.第二章研究投资者投资两种期望收益和风险都不相同的股票的最优投资组合问题,在影响股票价格的随机干扰源又相互关联的情况下,运用伊藤公式,采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.第三章研究投资股票和外汇存款的最优投资组合问题.这里允许股票卖空和从银行贷款,因此,分为了三种情况进行讨论.通过采取一种直接构造的方法,得到了最优投资组合及消费的显式解,并给出了值函数.第四章研究投资者在投资股票和外汇这样有风险的资产的同时,又将一部分资产投入到无风险的债券.针对幂效用函数,运用动态规划方法,得到了最优投资组合及消费的显式解,并给出了值函数.更多还原

【Abstract】 With the development and application of probability theory and stochastic process, studying optimal portfolio and consumption problems by using stochastic analysis methods has become one of the most common research methods in financial mathematics. It is re-garded as the real starting point of continuous-time portfolio theory that Merton formulated the continuous-time portfolio consumption model. By applying results of stochastic control theory to the portfolio problem he has able to obtain explicit solutions for some special ex-amples.In this paper, stochastic optimal portfolio problems for different models are studied. We choose constant relative risk aversion utility function, hyperbolic absolute risk aversion utility and power utility function respectively, using a direct method and dynamic program-ming approach, the optimal strategies are obtained, and the corresponding value functions are presented.This paper is organized as follows.In chapter 1, we introduce some background information, present development situation and research significance about the optimal portfolio problem. We systematically present some elementary knowledge about optimal portfolio problem.In chapter 2, we study the case in which the investor may choose two kinds of stocks with different expected returns and risks. Under the condition of correlated random interferences that affect the price of stocks, using Ito formula and a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.In chapter 3, we investigate the problem that the investor has two different investments. One is a stock, the other is a foreign exchange deposit. The investor is allowed to short-sell and loan from bank. Therefore, we divide into three kinds of case to discuss the problem. Using a direct method, the explicit optimal portfolio and consumption choice are obtained, and the corresponding value functions are presented.In chapter 4, we research the case that the investor invest part of his wealth in the risky assets and the rest in the risk-free assets. Using dynamic programming approach, the explicit optimal portfolio and consumption choice are obtained for the power utility function case, and the corresponding value functions are presented.更多还原