Rough paths and symmetric-Stratonovich integrals driven by singular covariance Gaussian processes.

may, 2024
Publication type:
Paper in peer-reviewed journals
Journal:
Bernoulli, vol. 30 (2), pp. 1197-1230
arXiv:
assets/images/icons/icon_arxiv.png 2206.06865
Keywords :
Rough paths; Stratonovich integrals.
Abstract:
We examine the relation between a stochastic version of the rough path 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 path 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 paths 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 $\mathbb{R}^2_+$ off diagonal.
BibTeX:
@article{Oha-Rus-2024,
    author={Alberto Ohashi and Francesco Russo },
    title={Rough paths and symmetric-Stratonovich integrals driven by 
           singular covariance Gaussian processes. },
    doi={10.3150/23-BEJ1629 },
    journal={Bernoulli },
    year={2024 },
    month={5},
    volume={30 (2) },
    pages={1197--1230},
}