case, we can recover Itô-type formulas for Gaussian processes, thus connecting our work to a substantial recent corpus e.g. [@np1998], [@amn2001], [@ccm2003], [@no2011] and [@hjt2013].

The bulk of the content in this paper can be found in the second-named author’s doctoral dissertation [@lim2016].

Preliminaries
=============

Rough path concepts and notation
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We briefly review the basic notation of rough paths theory used in this article; the standard references in this area [@lyons98], [@lq2003], [@fh2014] and [@fv2010b] can be consulted for more detail. We let $T^{n}\left(  \mathbb{R}^{d}\right)  $ denote the degree $n$ truncated tensor algebra $T^{n}\left(  \mathbb{R}^{d}\right) :=\mathbb{R}\oplus\mathbb{R}^{d}\oplus\cdots\oplus\left(  \mathbb{R}^{d}\right)^{\otimes n}$ equipped with addition and scalar multiplication as defined in the usual fashion. The truncated tensor product of $a=\left(  a^{0},a^{1},\ldots,a^{n}\right) $ (alternatively written as $a^{0}+a^{1}+\ldots+a^{n}$) and $b=\left(  b^{0},b^{1},\ldots,b^{n}\right)\in T^{n}\left(  \mathbb{R}^{d}\right)$ is defined by $$\begin{aligned}
a\otimes b:=\left(  c^{0},c^{1},\ldots,c^{n}\right), \qquad c^{k}=\sum
_{i=0}^{k}a^{i}\otimes b^{k-i},\quad\forall\,0\leq k\leq n,\end{aligned}$$ where here we abuse the notation by re-using the same symbol for the tensor product in $\mathbb{R}^{d}$. The unit element is $e=(1,0\ldots,0)$ and the tangent space of $T^{n}(\mathbb{R}^{d})$ at $e$ is denoted $A_{T}^{n}\left(\mathbb{R}^{d}\right)$. The exponential and logarithm maps, $\exp:A_{T}^{n}(\mathbb{R}^{d})\rightarrow T^{n}(\mathbb{R}^{d})$ and $\log: T^{n}(\mathbb{R}^{d})\rightarrow A_{T}^{n}(\mathbb{R}^{d}),$ are mutually inverse and defined by $$\begin{aligned}
\exp(a):=\sum_{i=0}^{n}\frac{a^{\otimes i}}{i!}, \qquad \log(a)=\sum_{i=1}^{n}(-1)^{i+1}\frac{(a-e)^{\otimes i}}{i}.\end{aligned}$$ The step-$n$ nilpotent 