On (p,q)-rough paths

We extend the work of T. Lyons [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] and T. Lyons and Z. Qian [T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr. Oxford Univ. Press, Oxford, 2002] to define integrals and solutions of differential equations along product of p and q rough paths, with 1/p+1/q>1. We use this to write an Itô formula at the level of rough paths, and to see that any rough path can always be interpreted as a product of a p-geometric rough path and a p/2-geometric rough path.