Stochastic Sewing Lemma Overview

• Stochastic sewing lemma is a key analytic result that constructs global processes from adapted random two-parameter increments with precise moment and conditional control.
• It extends rough path theory to stochastic settings, enabling coherent definitions of integrals for non-semimartingale noise in SDEs, SPDEs, and numerical schemes.
• Recent variants accommodate Banach space-valued processes, multiparameter fields, and discontinuities, broadening its application in modern stochastic analysis.

The stochastic sewing lemma is a core analytic result in stochastic analysis, generalizing the deterministic sewing lemma from rough path theory to the probabilistic setting. It provides sufficient conditions for reconstructing a global process from random, two-parameter increment data (typically adapted to a filtration), while controlling stochastic cancellation effects. The lemma and its variants underpin modern definitions of stochastic integrals in rough regimes, stability results for singular SDEs and SPDEs, error analyses in numerical approximations, and broad generalizations to Banach space-valued processes, multiple parameters, discontinuous controls, and probability measure flows.

1. Foundational Principles and Main Statement

The stochastic sewing lemma considers a two-parameter collection of random variables (or Banach space-valued processes) $A_{s,t}$, $0 \leq s < t \leq T$, adapted to a filtration, and seeks to construct—under suitable moment and conditional expectation estimates—a stochastic process $(\mathcal{A}_t)_{t \in [0,T]}$ such that for any partition $\pi$ of $[0,t]$, the Riemann sums $\sum_{[u,v]\in\pi}A_{u,v}$ converge in $L^m$ to $\mathcal{A}_t$ as the mesh goes to zero.

The principal regularity assumptions, as in (Lê, 2018), involve control of both:

• $$\|E[\delta A_{s,u,t}|\mathcal{F}_s]\|_{L^m} \leq T_1|t-s|^{1+\epsilon}$$
• $$\|A_{s,u,t} - E[A_{s,u,t}|\mathcal{F}_s]\|_{L^m} \leq T_2|t-s|^{1+\epsilon}$$, for some constants $T_1,T_2$, small $\epsilon>0$, and all $s < u < t$, where $\delta A_{s,u,t} = A_{s,t} - A_{s,u} - A_{u,t}$.

Under these, there exists a unique adapted process $(\mathcal{A}_t)$ with $\mathcal{A}_0=0$ such that $\mathcal{A}_t - \mathcal{A}_s$ can be approximated in $L^m$ by sums of $A_{u,v}$ over partitions of $[s,t]$, and the error is quantitatively controlled. The construction crucially leverages the increment's adaptation to the filtration and the improved (by a half) regularity requirement due to stochastic cancellation.

2. Extensions: Banach Spaces, Random Controls, and Generalized Regularity

Extending beyond finite-dimensional or scalar settings, (Lê, 2021) formulates the stochastic sewing lemma in Banach spaces $V$ with non-trivial martingale type $p \in (1,2]$. The conditions become:

• $$\|E_s[\delta A_{s,u,t}]\|_{L^n(V)} \leq \Gamma_1 w(s,t)^{1+\epsilon_1}$$
• $$\| \delta A_{s,u,t} - E_s[\delta A_{s,u,t}]\|_{L^m(V)} \leq \Gamma_2 w(s,t)^{1/p + \epsilon_2}$$ and the increments $A_{s,t}$ are required to be adapted to $\mathcal{F}_t$. The sewn process converges in $L^m(\Omega;V)$, with error bounds that reflect the Banach space martingale type.

Furthermore, recent progress (Butkovsky et al., 2023) demonstrated that one can relax uniform moment assumptions, as in the "Rosenthal-type stochastic sewing lemma". Instead of requiring all high moments to be bounded (as in the classical BDG-based version), bounds on the conditional second moment suffice, together with a mild higher-moment bound. Additional flexibility is introduced by allowing the control functions to be random and by formulating the lemma up to stopping times, vital in singular drift SDEs.

3. Multiparameter, Discontinuous, and Measure-Valued Generalizations

Multiparameter Case

A "multiparameter stochastic sewing lemma" is formulated for germ functions $A_{s,t}$ indexed over rectangles in $\mathbb{R}^d$ (Bechtold et al., 2023). Instead of simple additivity, multi-indexed delta operators and grid-based Riemann-type partitions are crucial. The main estimate typifies conditional expectations conditioned on multi-parameter filtrations, with moment bounds that decompose additively or multiplicatively in each coordinate's increment, enabling construction of Young-type integrals and local times for Gaussian fields on domains like $[0,1]^k$.

Discontinuous Controls and Jumps

Recent advances (Allan et al., 13 Aug 2024) allow the process increments $\Xi_{s,t}$ to be controlled by several, possibly discontinuous, control functions $w_{1,i},w_{2,i}$, each with their own exponents $$(\alpha_{1,i},\alpha_{2,i}),(\beta_{1,j},\beta_{2,j})$$. This enables the sewing lemma to handle increments with both roughness and jump-type discontinuities:

$$\|E_s[\delta \Xi_{s,u,t}]\|_{L^{qr}} \leq \sum_{i=1}^N w_{1,i}(s,t-)^{\alpha_{1,i}} w_{2,i}(s,t)^{\alpha_{2,i}}$$

$$\|\delta \Xi_{s,u,t}\|_{q,r,s} \leq \sum_{j=1}^M w_{1,j}(s,t-)^{\beta_{1,j}} w_{2,j}(s,t)^{\beta_{2,j}}$$

with exponents satisfying $\alpha_{1,i} + \alpha_{2,i} > 1$ and $\beta_{1,j} + \beta_{2,j} > 1/2$. The resulting process admits an error decomposition reflecting all control contributions, allowing robust integration theory for SDEs driven by noise with both rough and jump components, as well as a pathwise Itô formula encompassing jump-correction terms.

FloWs and Probability Measure Approximations

Measure-valued versions of the lemma (Alfonsi et al., 22 Oct 2024, Alfonsi et al., 2021) operate on operator families $\Theta_{s,t}: \mathcal{P}_p(H)\to\mathcal{P}_p(H)$ (probability measures on a Hilbert space). By exploiting probabilistic representations of the increment's difference and leveraging Wasserstein coupling, such sewing results construct a flow of laws/endomorphisms that can be applied directly to mean-field and McKean-Vlasov-type dynamics, even in the presence of law-dependent jumps and lack of pathwise uniqueness.

4. Applications: SDEs, SPDEs, Regularization by Noise, and Numerical Analysis

The stochastic sewing lemma is foundational for:

• Constructing Young- and rough-type integrals for random integrands and non-semimartingale noise (Brownian, fractional Brownian, or rough path lifts), including in Banach space settings (Lê, 2021).
• Providing novel definitions and error estimates for Euler–Maruyama and other numerical schemes for rough SDEs, including those with non-regular drifts, via careful sewing of local error increments (Butkovsky et al., 2019). The lemma yields strong convergence rates of order $n^{-\gamma+\delta}$ with sharp dependence of the exponent $\gamma$ on both drift regularity and noise Hurst parameter.
• Analysis and construction of mild solutions to singular SPDEs, in particular via mild stochastic sewing lemmas that incorporate the smoothing effects of evolution operators or semigroups (Li et al., 2021, Liang et al., 30 Jan 2024).
• Proving well-posedness, uniqueness, and stability of SDEs and SPDEs with rough, distributional, or measure-valued drift terms under optimal regularity conditions (Lê, 2018, Matsuda et al., 2022, Butkovsky et al., 2023), including pathwise uniqueness down to the optimal regime dictated by a balance of drift integrability and noise irregularity.
• Analyzing local times and regularization effects in multidimensional and multiparameter noise, giving additive regularity improvements as boundary effects in Gaussian fields (Bechtold et al., 2023).

A sample of key consequences:

• The sewing lemma enables a Doob–Meyer–type decomposition of the resulting process into martingale and compensator parts (Lê, 2018).
• The construction of weak solutions to law-dependent jump SDEs and propagation of chaos in stochastic particle systems by "lifting" the sewing construction to the probability measure level (Alfonsi et al., 22 Oct 2024, Alfonsi et al., 2021).

5. Technical Variants: Multiplicative, Mild, and Extended Sewing Lemmas

Multiplicative Sewing

The multiplicative sewing lemma (Gerasimovics et al., 2019, Hocquet et al., 2022) adapts the framework to operator-valued or non-commutative increments, constructing product integrals ("mild" propagators or semigroups) crucial for rough parabolic equations, even in infinite dimensions: $$Y(t,s) = \lim_{|\mathcal{P}|\to 0} p(t,t_n)\circ p(t_n,t_{n-1})\circ \cdots \circ p(t_1,s)$$ for $p$ approximately multiplicative and living in a suitable monoid.

Extensions to Fractional and Decorrelated Noise

The extended stochastic sewing lemma (Matsuda et al., 2022) builds in singular coefficients to account for conditional expectations taken at strictly earlier times ($v < s$), which is critical for processes (fractional Brownian motion, for instance) exhibiting asymptotic decorrelation. This makes it possible to prove convergence of stochastic integrals and occupation time discretizations under low regularity assumptions, with estimates containing factors of $(s-v)^{-\alpha}$ encoding the decorrelation rate.

Mild Sewing for SPDEs

The "mild stochastic sewing lemma" (Li et al., 2021, Liang et al., 30 Jan 2024) is adapted to convolution settings and incorporates the action of a semigroup or propagator, vital for defining integrals in infinite-dimensional evolution equations. The error control reflects both time increment and the propagator's smoothing indices.

6. Connections with Stochastic Reconstruction and Algebraic Structures

The stochastic sewing lemma is the analytic backbone of the stochastic reconstruction theorem (Kern, 2021), which extends Hairer's deterministically formulated reconstruction theorem from regularity structures to stochastic settings, gaining regularity via martingale cancellations. This provides analytic definitions of stochastic integrals (including against white noise) in distributional frameworks compatible with local coherence and improved conditional regularity.

Further, the lemma is harmonized with the deterministic, algebraic sewing on Hopf algebras (Broux et al., 2021), connecting it to rough path lifts, Lyons–Victoir extension theorems, and free actions on rough path spaces via shifted Hölder increments.

7. Summary of Representative Formulas

• Canonical sewing estimate (e.g., (Lê, 2018)):

$$\|A_{s,t} - (\mathcal{A}_t - \mathcal{A}_s)\|_{L^m} \leq C (T_1 + T_2)\, |t-s|^{1+\epsilon}$$

• Multiparameter, multidimensional sewing (Bechtold et al., 2023):

$$\mathbb{E}_s^\eta[ \delta_u^\theta A_{s,t} ] \lesssim \prod_{i\in\theta\setminus\eta}(t_i - s_i)^{1/2+\varepsilon_1} \prod_{i\in\eta}(t_i - s_i)^{1+\varepsilon_2}$$

• Extended (decorrelation by time gap) sewing (Matsuda et al., 2022):

$$\|E[\delta A_{s,u,t}|\mathcal{F}_v]\|_{L_m} \leq \Gamma_1\,(s-v)^{-\alpha} (t-s)^{\beta_1}, \quad v < s < u < t$$

• Jump/discontinuous control (multiple controls) sewing (Allan et al., 13 Aug 2024):

$$\|E_s[\delta I_{s,t}-\Xi_{s,t}]\| \leq C\,\sum_{i} w_{1,i}(s,t-)^{\alpha_{1,i}}w_{2,i}(s,t)^{\alpha_{2,i}}$$

8. Significance and Impact

The stochastic sewing lemma and its variants are now foundational tools in stochastic analysis, rough path theory, and the paper of singular SDEs/SPDEs. They provide the quantitative framework for constructing pathwise objects from locally controlled or approximately additive increments, rigorously exploiting probabilistic cancellations. Crucially, they enable definitions and stability proofs in numerous contexts for which classical (Itô) calculus is insufficient, including:

• SDEs/SPDEs with rough or distributional coefficients;
• Integration against fractional or non-semimartingale processes;
• Multidimensional and multi-parameter fields (e.g., Gaussian sheets);
• Numerical schemes for SDE approximations under minimal regularity;
• Regularization by noise and related "averaging" phenomena.

These lemmas bridge deterministic analytic structures with probabilistic and rough integration paradigms. Their technical flexibility—particularly in handling multiple parameters, discontinuities, Banach-valued processes, and measure evolutions—renders them central to ongoing advances in stochastic differential equations, stochastic numerics, and probabilistic regularity theory.