Journal Article10.1017/S0305004100070420
On homogeneous chaos
Nigel J. Cutland,Siu-Ah Ng +1 more
- 01 Sep 1991
- Vol. 110, Iss: 2, pp 353-363
TL;DR: In this paper, the authors discuss the Wiener-Ito chaos decomposition of an L 2 function <p over Wiener space, and are concerned in particular with the identification of the integrands /
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Abstract: This paper discusses the Wiener-Ito chaos decomposition of an L 2 function <p over Wiener space, and is concerned in particular with the identification of the integrands /„ in the chaos decomposition = &+ S [fndb™. n- l J First these are identified as Radon-Nikodym derivatives. Two elementary non-standard proofs of the Wiener-Ito chaos decomposition are given, based on Anderson's construction of Brownian motion and Ito integration. The non-standard framework then gives rigorous sense to Wiener's intuitive recipe i. db t where b, = —-, at which is meaningless in standard terms, but conveys the right idea. Finally, a Loebspace chaos decomposition result is used to give a generalization of a result of Nualart, Ustunel and Zakai.
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References
•Book
Nonstandard Analysis and its Applications
Nigel J. Cutland
- 30 Sep 1988
TL;DR: Non-standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology.
202
•Book
The Malliavin calculus
Denis Bell
- 01 Jan 1987
TL;DR: The Malliavin calculus of variations as discussed by the authors is an infinite dimensional differential calculus on the Wiener space, which was initiated by Malli-avin in 1974 to give a probabilistic proof of the Hörmander's theorem and its importance was immediately recognized.
188
Nonstandard Analysis and its Applications: AN INVITATION TO NONSTANDARD ANALYSIS
Ton Lindstrøm
- 01 Sep 1988
TL;DR: Nonstandard analysis has been used to obtain new results in such diverse fields as Banach spaces, differential equations, probability theory, algebraic number theory, economics, and mathematical physics just to mention a few as mentioned in this paper.
179