Cylindrical Wiener processes

TL;DR

This paper introduces a straightforward approach to defining cylindrical Wiener processes on Banach spaces and develops an associated stochastic integral without geometric constraints, connecting it to existing stochastic integrals.

Contribution

It provides a simple extension of real-valued Wiener processes to Banach spaces using cylindrical processes and defines a new stochastic integral applicable in this setting.

Findings

Defined cylindrical Wiener processes via cylindrical stochastic processes.

Introduced a stochastic integral with simple conditions on the integrand.

Connected the new integral to existing stochastic integrals in literature.

Abstract

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well-known stochastic integrals in literature.