## Lecture 12. p-rough paths

In this lecture, it is now time to harvest the fruits of the two previous lectures. This will allow us to finally define the notion of $p$-rough path and to construct the signature of such path.

A first result which is a consequence of the theorem proved in the previous lecture is the following continuity of the iterated iterated integrals with respect to a convenient topology. The proof uses very similar arguments to the previous two lectures, so we let it as an exercise to the student.

Theorem: Let $p \ge 1$, $K > 0$ and $x,y \in C^{1-var}([0,T],\mathbb{R}^d)$ such that
$\sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le 1,$
and
$\left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p+ \left( \sum_{j=1}^{[p]} \left\| \int dy^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right)^p \le K.$
Then there exists a constant $C \ge 0$ depending only on $p$ and $K$ such that for $k \ge 1$
$\left\| \int_{\Delta^k [0,\cdot]} dx^{\otimes k}- \int_{\Delta^k [0,\cdot]} dy^{\otimes k} \right\|_{p-var, [0,T]} \le \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \right) \frac{C^k}{\left( \frac{k}{p}\right)!}.$

This continuity result naturally leads to the following definition.

Definition: Let $p \ge 1$ and $x \in C^{p-var}([0,T],\mathbb{R}^d)$. We say that $x$ is a $p$-rough path if there exists a sequence $x_n \in C^{1-var}([0,T],\mathbb{R}^d)$ such that $x_n\to x$ in $p$-variation and such that for every $\varepsilon > 0$, there exists $N \ge 0$ such that for $m,n \ge N$,
$\sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le \varepsilon.$
The space of $p$-rough paths will be denoted $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$.

From the very definition, $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ is the closure of $C^{1-var}([0,T],\mathbb{R}^d)$ inside $C^{p-var}([0,T],\mathbb{R}^d)$ for the distance
$d_{\mathbf{\Omega}^p([0,T],\mathbb{R}^d)}(x,y)= \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} .$

If $x \in \mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ and $x_n \in C^{1-var}([0,T],\mathbb{R}^d)$ is such that $x_n\to x$ in $p$-variation and such that for every $\varepsilon > 0$, there exists $N \ge 0$ such that for $m,n \ge N$,
$\sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} \le \varepsilon,$
then we define $\int_{\Delta^k [s,t]} dx^{\otimes k}$ for $k \le p$ as the limit of the iterated integrals $\int_{\Delta^k [s,t]} dx_n^{\otimes k}$. However it is important to observe that $\int_{\Delta^k [s,t]} dx^{\otimes k}$ may then depend on the choice of the approximating sequence $x_n$. Once the integrals $\int_{\Delta^k [s,t]} dx^{\otimes k}$ are defined for $k \le p$, we can then use the previous theorem to construct all the iterated integrals $\int_{\Delta^k [s,t]} dx^{\otimes k}$ for $k > p$. It is then obvious that if $x,y \in \mathbf{\Omega}^p([0,T],\mathbb{R}^d)$, then
$1 + \sum_{k=1}^{[p]} \int_{\Delta^k [s,t]} dx^{\otimes k}=1 + \sum_{k=1}^{[p]} \int_{\Delta^k [s,t]} dy^{\otimes k}$
implies that
$1 + \sum_{k=1}^{+\infty } \int_{\Delta^k [s,t]} dx^{\otimes k}=1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,t]} dy^{\otimes k}.$
In other words the signature of a $p$-rough path is completely determinated by its truncated signature at order $[p]$:
$\mathfrak{S}_{[p]} (x)_{s,t} =1 + \sum_{k=1}^{[p]} \int_{\Delta^k [s,t]} dx^{\otimes k}.$
For this reason, it is natural to present a $p$-rough path by this truncated signature at order $[p]$ in order to stress that the choice of the approximating sequence to contruct the iterated integrals up to order $[p]$ has been made. This will be further explained in much more details when we will introduce the notion of geometric rough path over a rough path.

The following results are straightforward to obtain from the previous lectures by a limiting argument.

Lemma: Let $x \in \mathbf{\Omega}^p([0,T],\mathbb{R}^d)$, $p \ge 1$. For $0 \le s \le t \le u \le T$, and $n \ge 1$,
$\int_{\Delta^n [s,u]} dx^{\otimes n}=\sum_{k=0}^{n} \int_{\Delta^k [s,t]} dx^{\otimes k }\int_{\Delta^{n-k} [t,u]} dx^{\otimes (n-k) }.$

Theorem: Let $p \ge 1$. There exists a constant $C \ge 0$, depending only on $p$, such that for every $x \in\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ and $k \ge 1$,
$\left\| \int_{\Delta^k [s,t]} dx^{\otimes k} \right\| \le \frac{C^k}{\left( \frac{k}{p}\right)!} \left( \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}\right\|^{1/j}_{\frac{p}{j}-var, [s,t]} \right)^k, \quad 0 \le s \le t \le T.$

If $p \ge 2$, the space $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ is not a priori a Banach space (it is not a linear space) but it is a complete metric space for the distance
$d_{\mathbf{\Omega}^p([0,T],\mathbb{R}^d)}(x,y)= \sum_{j=1}^{[p]} \left\| \int dx^{\otimes j}- \int dy^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]} .$
The structure of $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ will be better understood in the next lectures, but let us remind that if $1 \le p < 2$, then $\mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ is the closure of $C^{1-var}([0,T],\mathbb{R}^d)$ inside $C^{p-var}([0,T],\mathbb{R}^d)$ for the variation distance it is therefore what we denoted $C^{0,p-var}([0,T],\mathbb{R}^d)$. As a corollary we deduce

Proposition: Let $1 \le p < 2$. Then $x \in \mathbf{\Omega}^p([0,T],\mathbb{R}^d)$ if and only if
$\lim_{\delta \to 0} \sup_{ \Pi \in \mathcal{D}[s,t], | \Pi | \le \delta } \sum_{k=0}^{n-1} \| x(t_{k+1}) -x(t_k) \|^p=0,$
where $\mathcal{D}[s,t]$ is the set of subdivisions of $[s,t]$. In particular, for $p < q < 2$,
$C^{q-var}([0,T],\mathbb{R}^d) \subset \mathbf{\Omega}^p([0,T],\mathbb{R}^d).$

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