  In this paper we construct a theory of stochastic integration of processes
with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and
$E$ is a UMD Banach space (i.e., a space in which martingale differences are
unconditional). The integrator is an $H$-cylindrical Brownian motion. Our
approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due
to Garling, which is combined with the theory of stochastic integration of
$\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors.
We obtain various characterizations of the stochastic integral and prove
versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities,
and the representation theorem for Brownian martingales.
