  We prove a stochastic Gronwall lemma of the following type: if $Z$ is an
adapted nonnegative continuous process which satisfies a linear integral
inequality with an added continuous local martingale $M$ and a process $H$ on
the right hand side, then for any $p \in (0,1)$ the $p$-th moment of the
supremum of $Z$ is bounded by a constant $\kappa_p$ (which does not depend on
$M$) times the $p$-th moment of the supremum of $H$. Our main tool is a
martingale inequality which is due to D. Burkholder. We provide an alternative
simple proof of the martingale inequality which provides an explicit numerical
value for the constant $c_p$ appearing in the inequality which is at most four
times as large as the optimal constant.
