  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.
