Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T08:21:43.588Z Has data issue: false hasContentIssue false

CONVOLUTION OF FUNCTIONALS OF DISCRETE-TIME NORMAL MARTINGALES

Published online by Cambridge University Press:  16 December 2011

QI HAN
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: [email protected])
CAISHI WANG*
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: [email protected], [email protected])
YULAN ZHOU
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: [email protected])
*
For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M=(M)n∈ℕ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. (‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl.373 (2011), 643–654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The authors are supported by National Natural Science Foundation of China (Grant No. 11061032) and Natural Science Foundation of Gansu Province (Grant No. 0710RJZA106). The second author is also supported partially by a grant from Northwest Normal University (Grant No. NWNU-KJCXGC-03-61).

References

[1]Conway, J. B., A Course in Functional Analysis, 2nd edn (Springer, New York, 1990).Google Scholar
[2]Émery, M., ‘A discrete approach to the chaotic representation property’, in: Séminaire de Probabilités, XXXV, Lecture Notes in Mathematics, 1755 (Springer, Berlin, 2001), pp. 123138.CrossRefGoogle Scholar
[3]Kallenberg, O., Foundations of Modern Probability (Springer, Berlin, 1997).Google Scholar
[4]Nourdin, I., Peccati, G. and Reinert, G., ‘Stein’s method and stochastic analysis of Rademacher functionals’, Electron. J. Probab. 15 (2010), 17031742.Google Scholar
[5]Privault, N., ‘Stochastic analysis of Bernoulli processes’, Probab. Surv. 5 (2008), 435483.CrossRefGoogle Scholar
[6]Wang, C. S., Chai, H. F. and Lu, Y. C., ‘Discrete-time quantum Bernoulli noises’, J. Math. Phys. 51 (2010), 053528.CrossRefGoogle Scholar
[7]Wang, C. S., Lu, Y. C. and Chai, H. F., ‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl. 373 (2011), 643654.CrossRefGoogle Scholar