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) ⊂ $S^ast$ (M)be the Gel’fand triple constructed from the functionals of M . As is known, there is no usual multiplication in $S^ast$ (M)since its elements are continuous linear functionals on S(M). However, by using the Fock transform, one can introduceconvolution in $S^ast$ (M), which suggests that one can try to introduce a type of integral of an $S^ast$ (M)-valued function withrespect to an $S^ast$ (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 $S^ast$ (M)-valued measures and examine their basic properties. Then, we define an integralof an $S^ast$ (M)-valued function with respect to an $S^ast$ (M)-valued measure and, among others, we establish a dominatedconvergence theorem for this integral. Finally, we also prove a Fubini type theorem for this integral.