On \((p,q)\)-rough paths. (English) Zbl 1097.60048

Summary: We extend the work of T. J. Lyons [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)] and T. Lyons and Z. Qian [“System control and rough paths” (2002; Zbl 1029.93001)] 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.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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