Skorohod and Stratonovich integrals for controlled processes

Given a continuous Gaussian process $x$ which gives rise to a $p$-geometric rough path for $pin (2,3)$, and a general continuous process $y$ controlled by $x$, under proper conditions we establish the relationship between the Skorohod integral $int_0^t y_s {mathrm{d}}^diamond x_s$ and the Stratonovich integral $int_0^t y_s {mathrm{d}} {mathbf x}_s$. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.