Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths

Abstract

Let C[0, T] denote an analogue of generalized Wiener space, the space of continuous real-valued functions on the interval [0, T].On the space C[0, T], we introduce a finite measure w α,β;ϕ and investigate its properties, where ϕ is an arbitrary finite measure on the Borel class of R. Using the measure w α,β;ϕ , we also introduce two measurable functions on C[0, T]; one of them is similar to the Itô integral and the other is similar to the Paley-Wiener-Zygmund integral. We will prove that if ϕ(R) = 1, then w α,β;ϕ is a probability measure with the mean function α and the variance function β, and the two measurable functions are reduced to the Paley-Wiener-Zygmund integral on the analogue of Wiener space C[0, T]. As an application of the integrals, we derive a generalized Paley-Wiener-Zygmund theorem which is useful to calculate generalized Wiener integrals on C[0, T]. Throughout this paper, we will recognize that the generalized Itô integral is more general than the generalized Paley-Wiener-Zygmund integral.