An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

TL;DR

This paper extends Lê's stochastic sewing lemma to handle more general conditional expectations, enabling new results in fractional stochastic calculus, including convergence of stochastic integrals, local time representations, and regularity conditions for SDEs driven by fractional Brownian motion.

Contribution

The paper introduces a generalized stochastic sewing lemma that incorporates asymptotic decorrelation, broadening its applicability to fractional stochastic calculus.

Findings

Proves convergence of stochastic integral approximations along fractional Brownian motion.

Provides new discretization-based representations of local times for fractional Brownian motion.

Improves regularity conditions for SDEs driven by fractional Brownian motion to ensure pathwise uniqueness.

Abstract

We give an extension of L\^e's stochastic sewing lemma [Electron. J. Probab. 25: 1 - 55, 2020]. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process $A$, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal {F}_s$. Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v < s$, and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal {F}_v$ by including a singularity in $(s-v)$. We provide three applications for which L\^e's stochastic sewing lemma seems to be insufficient.The first is to prove the convergence of It\^o or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.