Turkish Journal of Mathematics
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.
Keywords
Normal martingale, Fubini theorem, Bochner integral, vector measure
First Page
698
Last Page
711
Recommended Citation
JINSHU, CHEN
(2020)
"The Bochner-convolution integral for generalized functional-valued functions of discrete-time normal martingales,"
Turkish Journal of Mathematics: Vol. 44:
No.
3, Article 6.
https://doi.org/10.3906/mat-1912-21
Available at:
https://journals.tubitak.gov.tr/math/vol44/iss3/6
