Pathwise integration with respect to paths of finite quadratic variation

Journal article

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.

Authors: Ananova, A; Cont, R

• Alternative title: Intégration trajectorielle par rapport à des trajectoires de variation quadratique finie
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal de Mathématiques Pures et Appliquées
• Volume: 107
• Issue: 6
• Pages: 737-757
• Publication date: 2016-10-29
• Acceptance date: 2016-08-10
• DOI: 10.1016/j.matpur.2016.10.004
• EISSN: 0021-7824
• ISSN: 1776-3371
• Keywords: Ito isometry; Riemann sums; Pathwise integral; Quadratic variation; Stochastic integral