A stochastic sewing lemma and applications

TL;DR

This paper introduces a stochastic sewing lemma that extends Gubinelli's deterministic version, enabling convergence analysis of random Riemann sums and applications to stochastic differential equations with irregular drifts.

Contribution

It presents a stochastic version of the sewing lemma with improved regularity conditions and demonstrates its use in analyzing SDEs driven by Brownian and fractional Brownian motions.

Findings

Provides a sufficient condition for convergence of random Riemann sums

Establishes a Doob-Meyer-type decomposition for the limiting process

Applies the lemma to study SDEs with irregular drifts

Abstract

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.