Set-Valued Brown Motion And Its Properties

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.