Pathwise integration with respect to paths of finite quadratic variation

Abstract

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.

To appear in: Journal de Mathematiques Pures et Appliquees

References in corpus (5)

• Pathwise stochastic integrals for model free finance
• Local times for typical price paths and pathwise Tanaka formulas
• Weak approximation of martingale representations
• Doob--Meyer for rough paths
• Controlling Rough Paths

Cited by in corpus (9)

• Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity
• On pathwise quadratic variation for cadlag functions
• Bilinear equations in Hilbert space driven by paths of low regularity
• Comparison of path-dependent functionals of semimartingales
• Itô--Föllmer Calculus in Banach Spaces I: The Itô Formula
• Itô--Föllmer Calculus in Banach Spaces II: Transformations of Quadratic Variations
• Numerical method for model-free pricing of exotic derivatives using rough path signatures
• Fractional Ito calculus
• On isomorphism of the space of continuous functions with finite $p$-th variation along a partition sequence