Brownian representations of cylindrical continuous local martingales

TL;DR

This paper characterizes when cylindrical continuous local martingales can be represented as stochastic integrals with respect to cylindrical Brownian motions, and shows that a time change can always make them Brownian representable.

Contribution

It provides necessary and sufficient conditions for cylindrical martingales to be Brownian representable and introduces a time change technique for such representations.

Findings

Characterization of cylindrical martingales as stochastic integrals

Existence of a time change making any cylindrical martingale Brownian representable

Analysis of covariations generated by closed operators

Abstract

In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale $M$ there exists a time change $\tau$ such that $M\circ \tau$ is Brownian representable.