PATHWISE INTEGRATION AND CHANGE OF VARIABLE FORMULAS FOR CONTINUOUS PATHS WITH ARBITRARY REGULARITY

Abstract

We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of pth variation along a sequence of time partitions. For paths with finite pth variation along a sequence of time partitions, we derive a change of variable formula for p times continu-ously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an “isometry” formula in terms of pth order variation and obtain a “signal plus noise” decomposition for regular functionals of paths with strictly increasing pth variation. For less regular (Cp−1) functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of “reduced rough path”. We show that, while our integral coincides with a rough path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.