Pathwise Ito Calculus for Rough Paths

Abstract: We introduce path derivatives for controlled rough paths in the spirit of Dupire's functional Ito calculus. This allows us to study,  in a convenient manner, rough differential equations with time-dependent coefficients under minimal regularity assumptions (with respect to time). Consequently, we can establish existence and stability of pathwise solutions for a large class of SDEs on a universal canonical sample space. The results are useful for studying viscosity solutions of SPDEs.