An extension of Wiener integration with the use of operator theory

TL;DR

This paper extends Wiener integration by employing tensor products and operator theory to derive new approximation formulas and Fourier expansions for a broader class of stochastic integrals beyond Brownian motion.

Contribution

It introduces a novel extension of Wiener integration using tensor products and operator theory, overcoming limitations of classical stochastic integrals.

Findings

Derived an approximation formula for generalized stochastic integrals

Established a generalized Fourier expansion for these integrals

Discussed connections and estimates related to the extended approach

Abstract

With the use of tensor product of Hilbert space, and a diagonalization procedure from operator theory, we derive an approximation formula for a general class of stochastic integrals. Further we establish a generalized Fourier expansion for these stochastic integrals. In our extension, we circumvent some of the limitations of the more widely used stochastic integral due to Wiener and Ito, i.e., stochastic integration with respect to Brownian motion. Finally we discuss the connection between the two approaches, as well as a priori estimates and applications.