复Isonormal高斯过程及其导算子的可闭性

发布时间：2018-02-26 02:23

本文关键词： 复可分希尔伯特空间复Isonormal高斯过程复Hermite多项式复Malliavin导算子光滑随机变量 Wiener-It?chaos分解广义莱布尼茨法则　出处：《湖南科技大学》2016年硕士论文　论文类型：学位论文

【摘要】：以实可分的希尔伯特空间存在实Isonormal高斯过程为基础,在复可分的希尔伯特空间上建立了一个复Isonormal高斯过程。再从此复Isonormal高斯过程出发,运用复Hermite多项式定义了关于复Isonormal高斯过程的Wiener-It?chaos,并且通过复Hermite多项式正交性的性质,给出了平方可积空间的Wiener-It?chaos分解。接着在复Isonormal高斯过程上建立了光滑的随机变量函数类,进而在光滑的随机变量函数类中研究定义复Malliavin导算子。本论文的核心就是要说明复Malliavin导算子在光滑的随机变量函数类中是可闭化的。为此,先证明了实Isonormal高斯过程Malliavin导算子的可闭性,在这个研究过程中,发现并得到了Malliavin导算子的广义莱布尼茨法则,以及通过以Hermite多项式作为工具,得到散度算子的表示定理以及分部积分公式。作为一个应用,通过类比的方法,证明了复Isonormal高斯过程Malliavin导算子的可闭性。
[Abstract]:Based on the existence of real Isonormal Gao Si process in real separable Hilbert space, a complex Isonormal Gao Si process is established on the complex separable Hilbert space. The complex Hermite polynomials are used to define the Wiener-ITT process of the complex Isonormal Gao Si process. By using the property of orthogonality of complex Hermite polynomials, this paper gives the space of square integrable space Wiener-It? Chaos decomposition. Then in the complex Isonormal Gao Si process, the smooth class of random variable functions is established. The core of this paper is to show that the complex Malliavin derivative operator is closed in the class of smooth random variable functions. This paper first proves the closeness of the Malliavin derivative operator of real Isonormal Gao Si process. In this study, the generalized Leibniz rule of Malliavin derivative operator is found and obtained, and the Hermite polynomial is used as a tool. The representation theorem and integral formula of divergence operator are obtained. As an application, the closeness of Malliavin derivative operator of complex Isonormal Gao Si process is proved by analogy.
【学位授予单位】：湖南科技大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O177

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