Brownian representations of cylindrical continuous local martingales

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.