A Fubini type theorem for rough integration

TL;DR

This paper develops a Fubini type theorem for two-parameter rough integrals, extending the theory of controlled paths and establishing conditions under which multiple integrals coincide, with applications to the signature kernel in data science.

Contribution

It introduces a Fubini type theorem for two-parameter rough integrals, extending existing rough path theory to arbitrary regularity and different rough paths, with new continuity and maximal inequalities.

Findings

Proves the equivalence of different two-parameter rough integrals under certain conditions.

Extends Young-Towghi inequality to the rough path setting for two-parameter integrals.

Applies the theory to the signature kernel in data science.

Abstract

We develop the integration theory of two-parameter controlled paths $Y$ allowing us to define integrals of the form \begin{equation} \int_{[s,t] \times [u,v]} Y_{r,r'} \;d(X_{r}, X_{r'}) \end{equation} where $X$ is the geometric $p$-rough path that controls $Y$. This extends to arbitrary regularity the definition presented for $2\leq p<3$ in the recent paper of Hairer and Gerasimovi\v{c}s where it is used in the proof of a version of H\"{o}rmander's theorem for a class of SPDEs. We extend the Fubini type theorem of the same paper by showing that this two-parameter integral coincides with the two iterated one-parameter integrals \[ \int_{[s,t] \times [u,v]} Y_{r,r'} \;d(X_{r}, X_{r'}) = \int_{s}^{t} \int_{u}^{v} Y_{r,r'} \;dX_{r'} \;dX_{r'} = \int_{u}^{v} \int_{s}^{t} Y_{r,r'} \;dX_{r} \;dX_{r'}. \] A priori these three integrals have distinct definitions, and so this parallels the classical Fubini's theorem for product measures. By extending the two-parameter Young-Towghi inequality in this context, we derive a maximal inequality for the discrete integrals approximating the two-parameter integral. We also extend the analysis to consider integrals of the form \begin{equation*} \int_{[s,t] \times [u,v]} Y_{r,r'} \; d(X_{r}, \tilde{X}_{r'}) \end{equation*} for possibly different rough paths $X$ and $\tilde{X}$, and obtain the corresponding Fubini type theorem. We prove continuity estimates for these integrals in the appropriate rough path topologies. As an application we consider the signature kernel, which has recently emerged as a useful tool in data science, as an example of a two-parameter controlled rough path which also solves a two-parameter rough integral equation.