Shinzo Watanabe

A comparison theorem for solutions of stochastic differential equations and its applications

Introduction. Comparison theorem for solutions of stochastic differential equations was discussed by A.V. Skorohod [9] and T. Yamada [10], In §1, we will modify the main theorem of T. Yamada [10] so that it is more convenient for applications. As an application, we discuss in §2 some stochastic optimal control problem which was recently studied by V.E. Benes [1] using different methods. In §3, we obtain some comparison theorem for one-dimensional projection of a diffusion process. As an example of applications, we see that Hashiminskys test for explosion ([3], [7]) is obtained simply from a well known one-dimensional result.

Bessel diffusions as a one-parameter family of diffusion processes

O. Introduction By a Bessel diffusion process with index ~ (~ > 0), we mean a conservative onedimensional diffusion process on [0, o0) determined by the local generator

Fractional order Sobolev spaces on Wiener space

SummaryFractional order Sobolev spaces are introduced on an abstract Wiener space and Donskers delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Hölder continuity of a class of conditional expectations is obtained.

An Introduction to Malliavin's Calculus

Publisher Summary This chapter gives an introductory survey on the Malliavin calculus. It is an infinite dimensional differential calculus for functions on a Wiener space. This calculus is also called a stochastic calculus of variation for Wiener functionals because the functions on the Wiener space are functions of paths. The main aim of the Mailiavin calculus is to establish a differential calculus based on a Gaussian measure. The chapter explains that in the simple case of finite dimensional spaces where the proof is easily provided by the classical analysis, notions and formulas are aimed at extending in the general case of infinite dimensional spaces.

Lévy’s stochastic area formula and Brownian motion on compact Lie groups

Let (W, P) be the d-dimensional Wiener space: W be the space of continuous paths W = {w ∈ C([0, ∞) → R d )|w(0) = 0} and P be the standard d-dimensional Wiener measure on W. Then w = (w k(t)) k=1 d in W is a canonical realization of a d-dimensional Wiener process. Levy’s stochatic area is defined on the two-dimensional Wiener space by Ito’s stochastic integral as follows:

Donsker's δ-functions in the Malliavin calculus

S\left( t.\omega \right)=\frac{1}{2}\int{_{0}^{t}}{{w}^{1}}\left( s \right)d{{w}^{2}}\left( s \right)-{{w}^{2}}\left( s \right)d{{w}^{1}}\left( s \right)

Asymptotic Windings of Brownian Motion Paths on Riemann Surfaces

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A Fractional Calculus on Wiener Space

SummaryThe long time asymptotic behavior of the occupation times on a half line is studied for a class of one-dimensional diffusion processes whose excursion intervals have very heavy tail probability