【作者】 董方龙；

【导师】 赵联文；

【作者基本信息】 西南交通大学 ， 应用数学， 2003， 硕士

【摘要】 集值随机变量与集值随机过程作为一个新兴的研究领域，无论在理论上还是实际应用上面都不太成熟，不够完善，需要进一步的探索与开发。而作为随机过程的重要类型—布朗运动，在实际应用上具有很大的用处，本文的一个想法就是将集值与布朗运动结合起来，研究集值布朗运动，为集值随机过程的研究提供必要的理论基础。 本文首先给出了研究集值随机过程与集值随机变量所必需的知识，详细介绍了集值随机变量与它的可测选择之间的关系，得到了集值随机变量与集值随机过程的表示定理。 接着给出了一般的布朗运动的定义，讨论了一般的布朗运动与一般的鞅之间的关系，指出一般的布朗运动是一个一般的鞅，通过Banach空间的嵌入理论，给出了集值随机变量正态性的定义和集值随机过程的增量的定义；用过σ代数的独立性得到了集值随机过程的增量的独立性的定义，最后给出了集值布朗运动的定义并且类似于一般的布朗运动，给出了集值布朗运动的表示。 最后本文首先给出了集值鞅的定义，讨论了集值鞅的几个性质，给出了集值布朗运动与集值鞅之间的关系，证明了集值布朗运动是一个集值鞅。更多还原

【Abstract】 As a new and developing research field, Set-valued variable and Set-valued stochastic process is not perfect both in theory and practical application, and need to be further explored and developed. As an important aspect of stochastic process, Brown Motion has many practical applications. The thought of this article is to combine Set-valued stochastic process with Brown Motion, that is, to study Set-valued Brown motion. I hope that it can provide necessary theoretical foundation for Set-valued stochastic process.In this paper, firstly the basic knowledge of Set-valued variable and Set-valued stochastic process is proposed, and the relationship between Set-valued variable and its measurable selection is explicitly presented; and the representation of Set-valued variable and Set-valued stochastic process is given.Secondly, the concept of ordinary Brown Motion is given and the relationship between ordinary Brown Motion and ordinary martingale is discussed; furthermore, the conclusion that an ordinary Brown Motion is an ordinary martingale is proven. Through Banach Space Embedding Theory, the definition of the normality of Set-valued variable is given and the increment of Set-valued stochastic process is given. Through the independency of a -algebra, the independency of the increment of Set-valued stochastic process is given. At last, just like ordinary Brown Motion, the definition of Set-valued Brown Motion and itsrepresentation is given.Finally, the concept of Set-valued martingale is given, and some properties of Set-valued martingale is given, and the relationship between Set-valued martingale and Set-valued Brown Motion is given and proved: A Set-valued brown motion is a Set-valued martingale.更多还原