Selected Works of R.M. Dudley pp 443-444 | Cite as

# Wiener Functionals as Itô Integrals

Chapter

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## Abstract

Let
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

*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$$f(\omega )\, \equiv \,\int {_0^1 \varphi (t,\,\omega )d_t \,W(t,\,\omega )}$$

*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]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]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]Itô, K. and McKean, H. P., Jr. (1965).
*Diffusion Processes and their Sample Paths*. Springer-Verlag, New York.zbMATHGoogle Scholar

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