Stochastic Integrals and Gelfand Integration in Fr\'echet Spaces

TL;DR

This paper analyzes Gelfand integrals in Fréchet spaces, establishing key theorems and extending Skorohod and Malliavin calculus results, with applications to stochastic Volterra integration.

Contribution

It introduces a comprehensive framework for Gelfand integrals in Fréchet spaces, extending stochastic calculus tools and connecting them to Hida distributions and Volterra integrals.

Findings

Gelfand integral satisfies Vitali, dominated convergence, and Fubini theorems.

Skorohod integral expressed via Gelfand integral on Hida space.

Extended Malliavin calculus results and provided motivation for stochastic Volterra integration.

Abstract

We provide a detailed analysis of the Gelfand integral on Fr\'echet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida-Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.