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 $p$-th variation along a sequence of time partitions. For paths with finite $p$-th variation along a sequence of time partitions, we derive a change of variable formula for $p$ times continuously 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 $p$-th order variation and obtain a `signal plus noise' decomposition for regular functionals of paths with strictly increasing $p$-th variation. For less regular ($C^{p-1}$) 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.