is the conventional $k$-th iterated integral of the path $x$ over the interval $[s,t]$: $$\begin{aligned}
\mathbf{x}_{s,t}^{k} = \sum_{j_1, \ldots, j_k=1}^d \left( \int_{s<r_{1}<\cdots <r_{k}<t}\,\mathrm{d}x^{(j_1)}_{r_{1}}\otimes \cdots \otimes \,\mathrm{d}x^{(j_k)}_{r_{k}} \right) e_{j_1} \otimes \cdots \otimes e_{j_k}.\end{aligned}$$

For $p \geq 1$, the space of *geometric p-rough paths*, which we will denote by\
$\mathcal{C}^{0,p-var}\left( [0,T]; G^{\lfloor p \rfloor}\left( \mathbb{R}^{d}\right) \right)$, is defined to be the closure of $$\begin{aligned}
\mathcal{C}^{\infty} \left( [0, T]; G^{\lfloor p \rfloor} \left(\mathbb{R}^d\right) \right)
:= \left\{ f \in \mathcal{C}^{p-var} \left([0, T]; G^{\lfloor p \rfloor} \left(\mathbb{R}^d\right)\right) \: \Big| \: f \,  \mathrm{smooth} \right\}\end{aligned}$$ with respect to the topology given by the $p$-variation distance .

The notion of finite $p$-variation extends to (non-necessarily continuous) paths\
$f:[0,T]\rightarrow E$ taking values in a metric space $(E,d)$ via $$\begin{aligned}
\left\Vert f\right\Vert _{p-var;[0,T]}:=\sup_{\pi}\left(  \sum_{i}d(f_{t_{i}}, f_{t_{i+1}})_{E}^{p}\right)^{\frac{1}{p}}.\end{aligned}$$ Letting $V^{p-var}\left( [0,T]; E \right) := \{f \:|\: \left\Vert f\right\Vert_{p-var;[0,T]}<\infty\}$, we use $\mathcal{C}^{p-var}\left(  [0,T];E\right)$ and\
$\mathcal{C}_{pw}^{p-var}\left( [0, T]; E \right) $ to denote the subsets of $V^{p-var}\left( [0,T]; E\right) $ consisting of functions which are also, respectively, continuous functions and piecewise continuous. Given $f$ in\
$\mathcal{C}^{p-var}\left( [0,T];E\right)$, the function $\omega(s,t):=\left\Vert f\right\Vert _{p-var;[s,t]}^{p}$ defined on the simplex $\left\{ \left( s, t \right) \:|\: 0 \leq s\leq t\leq T\right\} $ is a *control*, by which we 