A Fubini type theorem for rough integration

Abstract

Jointly controlled paths as used in Hairer and Gerasimovičs (2019), are a class of two-parameter paths $Y$ controlled by a $p$-rough path $X$ for $2\le p<3$ in each time variable, and serve as a class of paths twice integrable with respect to $X$. We extend the notion of jointly controlled paths to two-parameter paths $Y$ controlled by $p$-rough and $\tilde{p}$-rough paths $X$ and $\tilde{X}$ (on finite dimensional spaces) for arbitrary $p$ and $\tilde{p}$, and develop the corresponding integration theory for this class of paths. In particular, we show that for paths $Y$ jointly controlled by $X$ and $\tilde{X}$, they are integrable with respect to $X$ and $\tilde{X}$, and moreover we prove a rough Fubini type theorem for the double rough integrals of $Y$ via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science applications, are jointly controlled paths.