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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