May 2024 Rough paths and symmetric-Stratonovich integrals driven by singular covariance Gaussian processes
Alberto Ohashi, Francesco Russo
Author Affiliations +
Bernoulli 30(2): 1197-1230 (May 2024). DOI: 10.3150/23-BEJ1629


We examine the relation between a stochastic version of the rough integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on R+2 off diagonal.

Funding Statement

This research was supported by MATH-AmSud 2018 (grant 88887.197425/2018-00) and Fundação de Apoio a Pesquisa do Destrito Federal (FAPDF grant 00193-00000229/2021-21). The research of the second named author was partially supported by the ANR-22-CE40-0015-01 (SDAIM).


Download Citation

Alberto Ohashi. Francesco Russo. "Rough paths and symmetric-Stratonovich integrals driven by singular covariance Gaussian processes." Bernoulli 30 (2) 1197 - 1230, May 2024.


Received: 1 August 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699550
Digital Object Identifier: 10.3150/23-BEJ1629

Keywords: Rough paths , Stratonovich integrals


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Vol.30 • No. 2 • May 2024
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