Lecture 19. Geometric rough paths

Posted on January 27, 2013 by Fabrice Baudoin

In this Lecture, the geometric concepts introduced in the previous lectures are now used to revisit the notion of p-rough path that was introduced before. We will see that using Carnot groups gives a perfect description of the space of p-rough paths through the notion of geometric rough path.

Definition: Let p \ge 1. An element x \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) is called a geometric p-rough path if there exists a sequence x_n \in C_0^{1-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) that converges to x in the p-variation distance. The space of geometric p-rough paths will be denoted by \mathbf{\Omega G}^p([0,T],\mathbb{R}^d).

To have it in mind, we recall the definition of a p-rough path.

Definition: Let p \ge 1 and x \in C_0^{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_0^{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 is denoted \mathbf{\Omega}^p([0,T],\mathbb{R}^d).

Our first goal is of course to relate the notion of geometric rough path to the notion of rough path.

Proposition: Let y \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) be a geometric p-rough path, then the projection of y onto \mathbb{R}^d is a p-rough path.

Proof: Let y \in C_0^{p-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) be a geometric p-rough path and let us consider a sequence y_n \in C_0^{1-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) that converges to y in the p-variation distance. Denote by x the projection of y onto \mathbb{R}^d and by x_n the projection of y_n. From a previous theorem y_n=S_{[p]}(x_n). It is clear that x_n converges to x in p-variation. So, we want to prove 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.
Let us now keep in mind that
d_{p-var; [s,t]}(y_n,y_m)=\left(\sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} d( y_n(t_k)^{-1}y_n(t_{k+1}) , y_m(t_k)^{-1}y_m(t_{k+1}))^{p}\right)^{1/p}.
and consider the control
\omega(s,t)=\left( \frac{ d_{p-var; [s,t]}(y_n,y_m)}{d_{p-var; [0,T]}(y_n,y_m) } \right)^p+\left( \frac{ d_{p-var; [s,t]}(0,y_m)}{d_{p-var; [0,T]}(0,y_m) } \right)^p.
We have
\left\| \int dx_n^{\otimes k}- \int dx_m^{\otimes k} \right\|_{\frac{p}{k}-var, [0,T]}
= \left(\sup_{ \Pi \in \mathcal{D}[0,T]} \sum_{j=0}^{n-1} \left\| \int_{\Delta^k [t_j,t_{j+1}]} dx_n^{\otimes k}- \int _{\Delta^k [t_j,t_{j+1}]} dx_m^{\otimes k} \right\|^{p/k}\right)^{k/p}
\le \left( \sup_{0 \le s \le t \le T} \frac{ \left\| \int_{\Delta^k [s,t]} dx_n^{\otimes k}- \int _{\Delta^k [s,t]} dx_m^{\otimes k} \right\|}{\omega(s,t)^{k/p} } \right) \omega(0,T)^{k/p}
From the ball-box estimate, there is a constant C such that for x,y \in \mathbb{G}_{[p]}(\mathbb{R}^d):
\| x-y \| \le C \max \{ d(x,y) \max \{ 1, d(0,x)^{N-1} \}, d(x,y)^N \}.
We deduce
\frac{ \left\| \int_{\Delta^k [s,t]} dx_n^{\otimes k}- \int _{\Delta^k [s,t]} dx_m^{\otimes k} \right\|}{\omega(s,t)^{k/p} }
\le C \max \left\{ d_{p-var; [0,T]}(y_n,y_m) \max \{ 1, d_{p-var; [0,T]}(0,y_m)^{N-1} \}, d_{p-var; [0,T]}(y_n,y_m)^N \right\}
and thus
\left\| \int dx_n^{\otimes k}- \int dx_m^{\otimes k} \right\|_{\frac{p}{k}-var, [0,T]} \le C' d_{p-var; [0,T]}(y_n,y_m)
This is the estimate we were looking for \square

Conversely, any p-rough path admits at least one lift as a geometric p-rough path.

Proposition: Let x \in C_0^{p-var} ([0,T], \mathbb{R}^d) be a p-rough path. There exists a geometric p-rough path y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d) such that the projection of y onto \mathbb{R}^d is x.

Proof: Consider a sequence x_n \in C_0^{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.
We claim that y_n=S_{[p]} (x_n) is a sequence that converges in p-variation to some y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d) such that the projection of y onto \mathbb{R}^d is x. Let us consider the control
\omega(s,t)= \left(\frac{ \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [s,t]}} { \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}} \right)^p+\left( \frac{ d_{p-var; [s,t]}(0,y_m)}{d_{p-var; [0,T]}(0,y_m) } \right)^p.
We have
d_{p-var; [0,T]}(y_n,y_m) \le \left( \sup_{0 \le s \le t \le T} \frac{ d \left( y_n(s)^{-1}y_n(t), y_m(s)^{-1}y_m(t) \right) }{\omega(s,t)^{1/p} } \right) \omega(0,T)^{1/p}
and argue as above to get, thanks to the ball-box theorem, an estimate like
d_{p-var; [0,T]}(y_n,y_m) \le C \left( \sum_{j=1}^{[p]} \left\| \int dx_n^{\otimes j}- \int dx_m^{\otimes j} \right\|^{1/j}_{\frac{p}{j}-var, [0,T]}\right)^{1/N}
\square

In general, we stress that there may be several geometric rough paths with the same projection onto \mathbb{R}^d. The following proposition is useful to prove that a given path is a geometric rough path.

Proposition: If q < p, then C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) \subset \mathbf{\Omega G}^p([0,T],\mathbb{R}^d).

Proof: As in Euclidean case, it is not difficult to prove that x \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d) if and only if
\lim_{\delta \to 0} \sup_{ \Pi \in \Delta[s,t], | \Pi | \le \delta } \sum_{k=0}^{n-1} d( x(t_k), x(t_{k+1}) )^p=0,
which is easy to check when x \in C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) \square

If y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d), then as we just saw, the projection x of y onto \mathbb{R}^d is a p-rough path and we can write
y(t)=1+\sum_{k=1}^{[p]} \int_{\Delta^k[0,t]} dx^{\otimes k}.
This is a convenient way to write geometric rough paths that we will often use in the sequel. For N \ge [p] we can then define the lift of y in \mathbf{\Omega G}^N([0,T],\mathbb{R}^d) as:
S_N(y)(t) =1+\sum_{k=1}^{N} \int_{\Delta^k[0,t]} dx^{\otimes k}.
The following result is then easy to prove by using the previous results.

Proposition: Let p \ge 1 and N \ge [p]. There exist constants C_1,C_2 > 0 such that for every y \in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d),
\| y \|_{p-var,[0,T]} \le \| S_{N} (y) \|_{p-var; [0,T]} \le C_2 \| y \|_{p-var,[0,T]}.

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