Kiyosi Itô

On a formula concerning stochastic differentials

In his previous paper [1] the author has stated a formula concering stochastic differentials with the outline of the proof. The aim of this paper is to show this formula in details in a little more general form (Theorem 6).

On the convergence of sums of independent Banach space valued random variables

The purpose of this paper is to discuss the convergence of sums of independent random variables with values in a separable real Banach space and to apply it to some problems on the convergence of the sample paths of stochastic processes. For the real random variables, we have a complete classical theory on the convergence of independent sums due to P. Levy, A. Khinchin and A. Kolmogorov. It can be extended to finite dimensional random variables without any change. In case the variables are infinite dimensional, there are several points which need special consideration. The difficulties come from the fact that bounded subsets of Banach space are not always conditionally compact. In Section 2 we will discuss some preliminary facts on Borel sets and probability measures in Banach space. In Section 3 we will extend P. Levys theorem. In Section 4 we will supplement P. Levys equivalent conditions with some other equivalent conditions, in case the random variables are symmetrically distributed. Here the infinite dimensionality will play an important role. The last section is devoted to applications.

Stochastic differential equations in a differentiable manifold

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Levy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

Infinite Dimensional Ornstein-Uhlenbeck Processes

Publisher Summary The chapter discusses the infinite dimensional Ornstein-Uhlenbeck processes. The chapter proves the infinite dimensional version of the fact that an Ornstein-Uhlenbeck process, a centered Gaussian, Markov, stationary and mean-continuous process { X t } satisfies the Langevin equation. An Orstein-Uhlenbeck process of linear random functionals is defined in the same way as in the 1-D case. There is a parallelism between the 1-D case and the infinite dimensional case, but an additional term, called the deterministic part is obtained. The chapter also discusses the continuous regular versions of the processes in consideration.

A Survey of Stochastic Differential Equations

Let us consider a system, dynamical,biological or economical, that is determined by a finite number of parameters:

Application to Markov Processes

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On stochastic differential equations

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Multiple Wiener Integral

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