CONVOLUTION ON FUNCTIONALS OF DISCRETE-TIME NORMAL MARTINGALE

Let $M=(M)_{n\in \mathbb{N}}$ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang \textit{et al.} (C.S. Wang, Y.C. Lu and H. F. Chai, 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 an interesting result, which shows that the convolution can be used to represent a certain family of conditional expectation operators associated with $M$. We also present an example of discrete-time normal martingale to show DOI: 10.1017/S000497271100309