<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.</p>
</abstract>

We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.

A Fourier Analysis Based New Look at Integration

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"A Fourier Analysis Based New Look at Integration"}

We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some...

A Fourier Analysis Based New Look at Integration

Published Online: Jul 26, 2023

Page range: 149 - 168

Received: Mar 29, 2023

Accepted: Jul 10, 2023

DOI: https://doi.org/10.2478/amsil-2023-0011

Keywords
Haar–Schauder series, Hölder spaces, sequence spaces, Paley-Littlewood packages, regularity, Young’s integral, rough path integral, para-control calculus, rough path analysis, controlled functions, para-controlled functions

© 2023 Peter Imkeller et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Fourier Analysis Based New Look at Integration

Published Online: Jul 26, 2023

Page range: 149 - 168

Received: Mar 29, 2023

Accepted: Jul 10, 2023

DOI: https://doi.org/10.2478/amsil-2023-0011

KeywordsHaar–Schauder series, Hölder spaces, sequence spaces, Paley-Littlewood packages, regularity, Young’s integral, rough path integral, para-control calculus, rough path analysis, controlled functions, para-controlled functions

© 2023 Peter Imkeller et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Haar–Schauder series, Hölder spaces, sequence spaces, Paley-Littlewood packages, regularity, Young’s integral, rough path integral, para-control calculus, rough path analysis, controlled functions, para-controlled functions