Lecture 12. p-rough paths

Posted on January 1, 2013 by Fabrice Baudoin

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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