  Using a Besov topology on spaces of modelled distributions in the framework
of Hairer's regularity structures, we prove the reconstruction theorem on these
Besov spaces with negative regularity. The Besov spaces of modelled
distributions are shown to be UMD Banach spaces and of martingale type $2$. As
a consequence, this gives access to a rich stochastic integration theory and to
existence and uniqueness results for mild solutions of semilinear stochastic
partial differential equations in these spaces of modelled distributions and
for distribution-valued SDEs. Furthermore, we provide a Fubini type theorem
allowing to interchange the order of stochastic integration and reconstruction.
