  We show that the weak disorder phase for the directed polymer model in a
bounded random environment is characterized by the integrability of the running
supremum $\sup_{n\in \mathbb N}W_n^\beta$ of the associated martingale
$(W_n^\beta)_{n\in \mathbb N}$. Using this characterization, we prove that
$(W_n^\beta)_{n\in \mathbb N}$ is $L^p$-bounded in the whole weak disorder
phase, for some $p>1$. The argument generalizes to non-negative martingales
with a certain product structure.
