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Turkish Journal of Mathematics

Authors

CHEN JINSHU

DOI

10.3906/mat-1912-21

Abstract

Let $M$ be a discrete-time normal martingale satisfying some mild conditions, $\mathcal{S}(M)\subset L^2(M)\subset\mathcal{S}^*(M)$ be the Gel'fand triple constructed from the functionals of $M$. As is known, there is no usual multiplication in $\mathcal{S}^*(M)$ since its elements are continuous linear functionals on $\mathcal{S}(M)$. However, by using the Fock transform, one can introduce convolution in $\mathcal{S}^*(M)$, which suggests that one can try to introduce a type of integral of an $\mathcal{S}^*(M)$-valued function with respect to an $\mathcal{S}^*(M)$-valued measure in the sense of convolution. In this paper, we just define such type of an integral. First, we introduce a class of $\mathcal{S}^*(M)$-valued measures and examine their basic properties. Then, we define an integral of an $\mathcal{S}^*(M)$-valued function with respect to an $\mathcal{S}^*(M)$-valued measure and, among others, we establish a dominated convergence theorem for this integral. Finally, we also prove a Fubini type theorem for this integral.

First Page

698

Last Page

711

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Mathematics Commons

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