Pathwise integration with respect to paths of finite quadratic variation

Anna Ananova (1) , Rama Cont (2)

1 Department of Mathematics [Imperial College London]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study a notion of pathwise integral, defined as the limit of non-anticipative Riemann sums, with respect to paths of finite quadratic variation. The construction allows to integrate 'gradient-type' integrands with respect to Hölder--continuous functions of Hölder index p< 1/2. We prove a pathwise isometry property for this integral, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands and obtain a pathwise 'signal plus noise' decomposition for a large class of irregular paths obtained through functional transformations of a reference path with non-vanishing quadratic variation.

Mots clés: Ito calculus; integration theory; rough path; Riemann sum

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

Anna Ananova, Rama Cont. Pathwise integration with respect to paths of finite quadratic variation. 2016. ⟨hal-01286515v1⟩