A stochastic sewing lemma and applications

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.