Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-23T08:59:41.282Z Has data issue: false hasContentIssue false
  • English
  • Français

On homogeneous chaos

Published online by Cambridge University Press:  24 October 2008

Nigel Cutland  and
Siu-Ah Ng
Nigel Cutland
Affiliation:
Department of Pure Mathematics, University of Hull, Hull HU6 7RX
Siu-Ah Ng
Affiliation:
Department of Pure Mathematics, University of Hull, Hull HU6 7RX
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decomposition

First these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 353 - 363
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Albeverio, S., Fenstad, J. E., Høegh-Krohn, R. and Lindstrøm, T.. Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Academic Press, 1986).Google Scholar
[2]Anderson, R. M.. A non-standard representation for Brownian motion and Itô integration. Israel J. Math. 25 (1976), 15–46.CrossRefGoogle Scholar
[3]Bell, D. R.. The Malliavin Calculus. Pitman Monographs and Surveys in Pure and Applied Math. vol. 34 (Longman, 1987).Google Scholar
[4]Cutland, N. J.. Nonstandard measure theory and its applications. Bull. London Math. Soc. 15 (1983), 529–589.CrossRefGoogle Scholar
[5]Cutland, N. J. (editor). Nonstandard Analysis and its Applications (Cambridge University Press, 1988).CrossRefGoogle Scholar
[6]Itô, K.. Multiple Wiener integral. J. Math. Soc. Japan 3 (1951), 157–169.CrossRefGoogle Scholar
[7]Lindstrom, T.. An invitation to nonstandard analysis, in [5], pp. 1–105.CrossRefGoogle Scholar
[8]McKean, H. P.. Geometry of differential space. Ann. Probab. 1 (1973), 197–206.CrossRefGoogle Scholar
[9]Nualart, D., Ustunel, A. S. and Zakai, M.. On the moments of a multiple Wiener-Ito integral and the space induced by the polynomials of the integral. Stochastics 25 (1988), 233–240.CrossRefGoogle Scholar
[10]Stroock, D. W.. The Malliavin calculus and its applications. In Stochastic Integrals, Lecture Notes in Math. vol. 851 (Springer-Verlag, 1987), pp. 394–432.CrossRefGoogle Scholar
[11]Stroock, D. W.. Homogeneous chaos revisited. In Séminaire de Probabilités XXI, Lecture Notes in Math. vol. 1247 (Springer-Verlag, 1987), pp. 1–7.CrossRefGoogle Scholar
[12]Wiener, N.. The homogeneous chaos. Amer. J. Math. 60 (1938), 897–936.CrossRefGoogle Scholar