Jointly controlled paths as used in Gerasimovics and Hairer (2019), are a class of two-parameter paths Y controlled by a p-rough path X for 2 ≤p < 3 in each time variable, and serve as a class of paths twice integrable with respect to X. We extend the notion of jointly controlled paths to two-parameter paths Y controlled by p-rough and Qp-rough paths X and QX (on finite dimensional spaces) for arbitrary p and Qp, and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and QX , they are integrable with respect to X and QX, and moreover we prove a rough Fubini type theorem for the double rough integrals of Y via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science
applications, are jointly controlled paths.