Advertisement

Wiener Functionals as Itô Integrals

  • R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Let W(t, ω) be a standard Wiener process, W t W(t) ≡ W(t, ∙). A function φ(t, ω) is called nonanticipating iff for all t ^ 0, φ(t, ∙) is measurable with respect to {W s : 0 % s % t}. The Itô stochastic integral
$$f(\omega )\, \equiv \,\int {_0^1 \varphi (t,\,\omega )d_t \,W(t,\,\omega )}$$
is defined for any jointly measurable, nonanticipating φ such that for almost all \(\Omega, \int_0^1 \Phi^2(t, \Omega)dt < \infty\) (Gikhman and skorokhod (1968), Chapter 1, Section 2). It is known that it is defined for any jointly measurable, nonanticipating φ such that for almost all \(E \int_0^1 \Phi^2(t, \Phi) dt < \infty, {\rm then} Ef = 0 {\rm and} Ef^2 < \infty.\). Representation of an arbitrary measurable f as a stochastic integral was stated, but later retracted, by J.M.C. Clark (1970, 1971).

Key words and phrases

Stochastic integral Wiener process 

References

  1. [1]
    Clark, J. M. C. (1970, 1971). The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Statist. 41 1282–1295; correction, ibid. 42 1778.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Gikhman, I. I. and Skorokhod, A. V. (1968). Stochastic Differential Eqautions. Naukova Dumka, Kiev, (in Russian); Akademie-Verlag, Berlin, 1971 (in German); Springer-Verlag, New York, 1972 (in English).Google Scholar
  3. [3]
    Itô, K. and McKean, H. P., Jr. (1965). Diffusion Processes and their Sample Paths. Springer-Verlag, New York.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Massachusetts Intstitute of TechnologyCambridgeUS

Personalised recommendations