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We examine the relation between a stochastic version of the rough pathintegral with the symmetric-Stratonovich integral in the sense ofregularization. Under mild regularity conditions in the sense of Malliavincalculus, we establish equality between stochastic rough path andsymmetric-Stratonovich integrals driven by a class of Gaussian processes. As aby-product, we show that solutions of multi-dimensional rough differentialequations driven by a large class of Gaussian rough paths they are actuallysolutions to Stratonovich stochastic differential equations. We obtain almostsure convergence rates of the first-order Stratonovich scheme to rough pathsintegrals in the sense of Gubinelli. In case the time-increment of theMalliavin derivative of the integrands is regular enough, the rates areessentially sharp. The framework applies to a large class of Gaussian processeswhose the second-order derivative of the covariance function is a sigma-finitenon-positive measure on ${\mathbb R}^2$ + off diagonal.

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