Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

Integration trajectorielle et formules de changement de variables pour des fonctions continues de regularite arbitraire

Rama Cont (1, 2) , Nicolas Perkowski (3)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 Department of Mathematics [Imperial College London]
3 Institut für Mathematik [Humboldt]

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 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.

Mots clés: functional calculus; rough paths; integration

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]

Rama Cont, Nicolas Perkowski. Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity. Transactions of the American Mathematical Society, 2019, 6 (161-186), ⟨10.1090/btran/34⟩. ⟨hal-01742614v2⟩