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CONVOLUTION OF FUNCTIONALS OF DISCRETE-TIME NORMAL MARTINGALES

Part of: Stochastic analysis

Published online by Cambridge University Press:  16 December 2011

QI HAN ,
CAISHI WANG  and
YULAN ZHOU
QI HAN
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: hanqi1978@nwnu.edu.cn)
CAISHI WANG*
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: wangcs@nwnu.edu.cn, cswangnwnu@163.com)
YULAN ZHOU
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: zhouylw123@163.com)
*
✉For correspondence; e-mail: cswangnwnu@163.com
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Abstract

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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.

Keywords

normal martingalefunctionalconvolution

MSC classification

Secondary: 60H40: White noise theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 224 - 231
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

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