The Bochner-convolution integral for generalized functional-valued functions of discrete-time normal martingales

Abstract

Let M be a discrete-time normal martingale satisfying some mild conditions, S M ⊂ L 2 M ⊂ S∗ M be the Gel’fand triple constructed from the functionals of M . As is known, there is no usual multiplication in S ∗ M since its elements are continuous linear functionals on S M . However, by using the Fock transform, one can introduce convolution in S ∗ M , which suggests that one can try to introduce a type of integral of an S ∗ M -valued function with respect to an 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 S ∗ M -valued measures and examine their basic properties. Then, we define an integral of an S ∗ M -valued function with respect to an 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.