Skorohod and Stratonovich integrals for controlled processes

Given a continuous Gaussian process $x$ which gives rise to a $p$-geometric rough path for $p\in (2,3)$, 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.