  We introduce a stochastic version of Gubinelli's sewing lemma, providing a
sufficient condition for the convergence in moments of some random Riemann
sums. Compared with the deterministic sewing lemma, adaptiveness is required
and the regularity restriction is improved by a half. The limiting process
exhibits a Doob-Meyer-type decomposition. Relations with It\^o calculus are
established. To illustrate further potential applications, we use the
stochastic sewing lemma in studying stochastic differential equations driven by
Brownian motions or fractional Brownian motions with irregulardrifts.
