Pathwise integration with respect to paths of finite quadratic variation

Integration trajectorielle par rapport aux trajectoires à variation quadratique finie

Anna Ananova (1) , Rama Cont (2)

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

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, 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. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero 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-01286515v3⟩