集值逆上鞅的收敛定理及应用

研究了取值于Ｂａｎａｃｈ空间的集值逆上鞅的收敛性，给出了集值逆上鞅在集列Ｗｉｊｓｍａｎ收敛，弱收敛及Ｋｕｒａｔｏｗｓｋｉ－Ｍｏｓｃｏ收敛意义下的收敛定理，并给出了它们在连续参数集值鞅中的应用。     

Martingales,the smoothing index and ergodic theory

A classical theorem of Meyer Jerison which shows that the convergence in the pointwise ergodic theorem is equivalent to the convergence of an associated martingale is expanded to a conditional setting. An equiconvergence theorem of the type established for martingales by N.F.G. Martin and E. Boylan is established in the ergodic case for an ergodic, non-invertible, measure-preserving transformation.     

On functional central limit theorems for certain continuous time parameter stochastic processes

Weak invariance principles for certain continuous time parameter stochastic processes (including martingales and reverse martingales) are considered. Weak convergence in the sup-norm metric is also studied.     

B值渐近鞅的局部收敛性

Ｂ值渐近鞅是Ｂ值鞅的重要推广,它保持了鞅的一些是一性质,然后对Ｂ值渐近鞅的局部收敛性很少有文献论及。本文利用Ｂ值渐近鞅的Ｄｏｏｂ分解,对Ｂ值渐近鞅的局部收敛性作些探讨,得到了Ｂ值渐近鞅局部收敛性的几个结果,它们是鞅的有关结论的推广与改进。     

Stochastic Integrals of Continuous Local Martingales,II

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n-dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n-dimensional time change for reducing every n-dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.     

Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

Rates of convergence in the functional CLT for multidimensional continuous time martingales

New rates of convergence in the multidimensional functional CLT are given by means of the Prokhorov's distance between a brownian motion and a continuous time martingale, with no further assumption than square integrability. The results are completely and simply expressed with distances of predictable characteristics which naturally occur in various statements of CLT for martingales.     

Stochastic integration for operator valued processes on hilbert spaces and on nuclear spaces

The representation of a nuclear space valued square integrable martingale by means of another nuclear space valued square integrable martingale is given in terms of stochastic inegrals of operator valued processes. The construction of the stochastic integral goes through that of operator valued processes on Hilbert spaces. A new approach is given for the Hilbertian case, so that only the integration of Hilbert-Schmidt operator valued processes is needed to represent square integrable martingales     

可积鞅测度的弱收敛(英语)

本文引入了可积鞅测度弱收敛的概念,并给出了可积鞅测弱收敛的一系列条件     

Stochastic Integrals of Continuous Local Martingales,I

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǒ integral of X. Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n-dimensional case, and, in particular, to the n-dimensional time change of continuous local martingales with orthogonal components.