Stochastic integration with respect to the cylindrical Wiener process via regularization

TL;DR

This paper introduces a new stochastic integral with respect to the cylindrical Wiener process using regularization, extending classical integrals, and applies it to solve a stochastic PDE with anticipating initial data.

Contribution

It develops a forward integral approach for cylindrical Wiener processes, expanding stochastic calculus tools for infinite-dimensional processes.

Findings

Established a new stochastic integral for cylindrical Wiener processes.

Proved existence of solutions for a class of stochastic PDEs with anticipating initial data.

Extended classical stochastic calculus to infinite-dimensional settings.

Abstract

Following the ideas of F. Russo and P. Vallois we use the notion of forward integral to introduce a new stochastic integral respect to the cylindrical Winer process. This integral is an extension of the classical integral. As an application, we prove existence of solution of a parabolic stochastic differential partial equation with anticipating stochastic initial date.