Rough paths and symmetric-Stratonovich integrals driven by singular covariance gaussian processes
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2022
- DOI:
- 10.48550/arXiv.2206.06865
- arXiv:
- arXiv:2206.06865
- Bibcode:
- 2022arXiv220606865O
- Keywords:
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- Mathematics - Probability
- E-Print:
- Bernoulli, In press