Research and Lecture notes

Lecture 29. Stochastic differential equations as rough differential equations

Posted on April 6, 2013 by Fabrice Baudoin

Based on the results of the previous Lecture, it should come as no surprise that differential equations driven by the Brownian rough path should correspond to Stratonovitch differential equations. In this Lecture, we prove that it is indeed the case. Let us first remind to the reader the following basic result about existence and uniqueness for solutions of stochastic differential equations.

Let $(B_t)_{t \geq 0}=(B^1_t,...,B^d_t)_{t \geq 0}$ be a $d$ -dimensional Brownian motion defined on some filtered probability space $\left( \Omega , (\mathcal{F}_t)_{t \geq 0} , \mathbb{P} \right)$ that satisfies the usual conditions.

Theorem: Assume that $V_1,\cdots,V_d$ are $C^2$ vector fields with bounded derivatives up to order 2. Let $x_0 \in \mathbb{R}^n$ . On $\left( \Omega , (\mathcal{F}_t)_{t \geq 0} , \mathbb{P} \right)$ , there exists a unique continuous and adapted process $(X_t)_{t \geq 0}$ such that for $t \geq 0$ ,
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \circ dB^i_s.$

Thanks to Ito’s formula the corresponding Ito’s formulation is
$X_t=x_0 + \frac{1}{2} \sum_{i=1}^d \int_0^t \nabla_{V_i} V_i (X_s) ds +\sum_{i=1}^d \int_0^t V_i (X_s ) dB^i_s,$
where for $1 \leq i \leq d$ , $\nabla_{V_i} V_i$ is the vector field given by
$\nabla_{V_i} V_i (x)=V_i^2 \mathbf{I} (x)= \sum_{j=1}^n \left( \sum_{k=1}^n v_i^k (x) \frac{\partial v^j_i}{\partial x_k}(x)\right)\frac{\partial}{\partial x_j}, \text{ }x \in \mathbb{R}^n.$

The main result of the Lecture is the following:

Theorem: Let $\gamma > 2$ and let $V_1,\cdots,V_d$ be $\gamma$ -Lipschitz vector fields on $\mathbb{R}^n$ . Let $x_0 \in \mathbb{R}^n$ . The solution of the rough differential equation
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \ dB^i_s,$
is the solution of the Stratonovitch differential equation:
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \circ dB^i_s.$

Proof: Let us work on a fixed interval $[0,T]$ and consider a sequence $D_n$ of subdivisions of $[0,T]$ such that $D_{n+1} \subset D_n$ and whose mesh goes to 0 when $n \to +\infty$ . As in the previous lectures, we denote by $B^n$ the piecewise linear process which is obtained from $B$ by interpolation along the subdivision $D_n$ , that is for $t_i^n \le t \le t_{i+1}^n$ ,
$B^n_t= \frac{t_{i+1}^n -t}{ t_{i+1}^n-t_i^n} B_{t^n_i} + \frac{t-t_i^n}{ t_{i+1}^n-t_i^n} B_{t^n_{i+1}}.$
Let us then consider the process $X_n$ that solves the equation
$X^n_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X^n_s) \ dB^{i,n}_s,$
and the process $\tilde{X}^n$ , which is piecewise linear and such that
$\tilde{X}^n_{t^n_{k+1}}=\tilde{X}^n_{t^n_{k}}+\sum_{i=i}^d V_i ( X^n_{t_k^n}) ( B^i_{t^n_{k+1}}- B^i_{t^n_{k}})+\frac{1}{2} \sum_{i=1}^d V_i^2 \mathbf{I} (X^n_{t_k^n})(t_{k+1}^n -t_k^n).$
We can write
$X_{t_{k+1}^n} - \tilde{X}_{t_{k+1}^n} = \sum_{\nu=0}^k ( X_{t_{\nu+1}^n} - X_{t_{\nu}^n} )- (\tilde{X}_{t_{\nu+1}^n} - \tilde{X}_{t_{\nu}^n} ).$
Now,
$( X_{t_{\nu+1}^n} - X_{t_{\nu}^n} )- ( \tilde{X}_{t_{\nu+1}^n} - \tilde{X}_{t_{\nu}^n} )$
$= ( X_{t_{\nu+1}^n} - X_{t_{\nu}^n} )- \sum_{i=i}^d V_i ( X^n_{t_\nu^n}) ( B^i_{t^n_{\nu+1}}- B^i_{t^n_{\nu}}) -\frac{1}{2} \sum_{i=1}^d V_i^2 \mathbf{I} (X^n_{t_\nu^n})(t_{\nu+1}^n -t_\nu^n).$
From Davie’s estimate, we have, with $2 < p < \gamma$ ,
$\left\|( X_{t_{\nu+1}^n} - X_{t_{\nu}^n} )- \sum_{i=i}^d V_i ( X^n_{t_\nu^n}) ( B^i_{t^n_{\nu+1}}- B^i_{t^n_{\nu}}) -\sum_{i,j=1}^d (V_i V_j \mathbf{I}) (X^n_{t_\nu^n}) \int_{t_\nu^n}^{t_{\nu+1}^n} (B^{n,i}_u-B^{n,i}_{t_\nu^n})dB^{n,j}_u \right\|$
$\le C \| V \|_{Lip^{\gamma-1}} \| S_2(B^n) \|^\gamma_{p-var, [t^n_{\nu}, t^n_{\nu+1}]}$
$\le C \| V \|_{Lip^{\gamma-1}} \| B^n \|^\gamma_{p-var, [t^n_{\nu}, t^n_{\nu+1}]}$
$\le C' \| V \|_{Lip^{\gamma-1}} \| B \|^\gamma_{p-var, [t^n_{\nu}, t^n_{\nu+1}]}.$
We deduce that, almost surely when $n \to \infty$ ,
$\sum_{\nu=0}^k \left\|( X_{t_{\nu+1}^n} - X_{t_{\nu}^n} )- \sum_{i=i}^d V_i ( X^n_{t_\nu^n}) ( B^i_{t^n_{\nu+1}}- B^i_{t^n_{\nu}}) -\sum_{i,j=1}^d (V_i V_j \mathbf{I}) (X^n_{t_\nu^n}) \int_{t_\nu^n}^{t_{\nu+1}} (B^{n,i}_u-B^{n,i}_{t_\nu^n})dB^{n,j}_u \right\| \to 0.$
On the other hand,
$\left\| \sum_{i,j=1}^d (V_i V_j \mathbf{I}) (X^n_{t_\nu^n}) \int_{t_\nu^n}^{t_{\nu+1}^n} (B^{n,i}_u-B^{n,i}_{t_\nu^n})dB^{n,j}_u -\frac{1}{2} \sum_{i=1}^d V_i^2 \mathbf{I} (X^n_{t_\nu^n})(t_{\nu+1}^n -t_\nu^n)\right\|$
$\le\| V \|_{Lip^{\gamma}} \sum_{i,j=1}^d \left| \int_{t_\nu^n}^{t_{\nu+1}^n} (B^{n,i}_u-B^{n,i}_{t_\nu^n})dB^{n,j}_u - \frac{1}{2} \delta_{ij} (t_{\nu+1}^n -t_\nu^n) \right|$
$\le \frac{1}{2} \| V \|_{Lip^{\gamma}} \sum_{i,j=1}^d \left| (B^{n,i}_{t_{\nu+1}^n}-B^{n,i}_{t_\nu^n}) (B^{n,j}_{t_{\nu+1}^n}-B^{n,j}_{t_\nu^n})- \delta_{ij} (t_{\nu+1}^n -t_\nu^n) \right|$
We deduce that in probability,
$\sum_{\nu=0}^k \left\| \sum_{i,j=1}^d (V_i V_j \mathbf{I}) (X^n_{t_\nu^n}) \int_{t_\nu^n}^{t_{\nu+1}^n} (B^{n,i}_u-B^{n,i}_{t_\nu^n})dB^{n,j}_u -\frac{1}{2} \sum_{i=1}^d V_i^2 \mathbf{I} (X^n_{t_\nu^n})(t_{\nu+1}^n -t_\nu^n)\right\| \to 0.$
We conclude that in probability,
$X_{t_{k+1}^n} - \tilde{X}_{t_{k+1}^n} \to 0.$
Up to an extraction of subsequence, we can assume that almost surely
$X_{t_{k+1}^n} - \tilde{X}_{t_{k+1}^n} \to 0.$
We now know that from the Lyons’ continuity theorem, almost surely $X_t^n \to X_t$ where $(X_t)_{t \in [0,T]}$ is the solution of the rough differential equation
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \ dB^i_s.$
Thus almost surely, we have that $\tilde{X}_t^n \to X_t$ . On the othe hand, by definition, we have
$\tilde{X}^n_{t^n_{k+1}}=\tilde{X}^n_{t^n_{k}}+\sum_{i=i}^d V_i ( X^n_{t_k^n}) ( B^i_{t^n_{k+1}}- B^i_{t^n_{k}})+\frac{1}{2} \sum_{i=1}^d V_i^2 \mathbf{I} (X^n_{t_k^n})(t_{k+1}^n -t_k^n),$
which easily implies that $\tilde{X}^n$ converges in probability to $x_0+\sum_{i=i}^d \int_0^t V_i (X_s)\circ dB^i_s$ . This proves that
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \circ dB^i_s$
$\square$

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Lecture 28. Signature of the Brownian rough path

Posted on March 31, 2013 by Fabrice Baudoin

Since a $d$ -dimensional Brownian motion $(B_t)_{t \ge 0}$ is a $p$ -rough path for $p > 2$ , we know how to give a sense to the signature of the Brownian motion.
In particular, the iterated integrals at any order of the Brownian motion are well defined using rough path theory. It turns out that these iterated integrals do not coincide with iterated Ito’s integrals but with iterated Stratonovitch integrals.
We start with some reminders about Stratonovitch integration. Let $(B_t)_{t \ge 0}$ be a one dimensional Brownian motion defined on a filtered probability space $(\Omega, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ . Let $(\Theta_t)_{0 \le t \le T}$ be a $\mathcal{F}$ adapted process such that $\mathbb{E} \left( \int_0^T \Theta_s^2 ds \right)< +\infty$ . The Stratonovitch integral of $\Theta$ against $B$ can be defined as the limit in probability of the sums
$\sum_{k=0}^{n-1} \frac{\Theta_{t_{k+1}^n} +\Theta_{t_{k}^n}}{2} (B_{t_{k+1}^n}-B_{t_{k}^n}),$
where $0=t_0^n \le t_1^n \le \cdots \le t_n^n=T$ is a sequence of subdivisions whose mesh goes to 0. This limit is denoted $\int_0^T \Theta_s \circ dB_s$ and does not depend on the choice of the subdivision. It is an easy exercise to see that the relation between Ito’s integral and Stratonovitch’s is given by:
$\int_0^T \Theta_s \circ dB_s=\int_0^T \Theta_s dB_s+\frac{1}{2} \langle \Theta, B \rangle_T,$
where $\langle \Theta, B \rangle_T$ is the quadratic covariation between $\Theta$ and $B$ .
If $(B_t)_{t \ge 0}$ is $d$ dimensional Brownian motion, we can then inductively define the iterated Stratonovitch integrals $\int_{0 \leq t_1 \leq ... \leq t_k \leq t} \circ dB^{i_1}_{t_1} \cdots \circ dB^{i_k}_{t_k}$ . The next theorem proves that the signature of the Brownian rough path is given by multiple Stratonovitch integrals.

Theorem: If $(B_t)_{t \ge 0}$ is a $d$ -dimensional Brownian motion, the signature of $B$ as a rough path is the formal series:
$\mathfrak{S} (B)_t = 1+ \sum_{k=1}^{+\infty} \int_{\Delta^k[0,t]} \circ dB ^{\otimes k}$
$=1 + \sum_{k=1}^{+\infty} \sum_{I \in \{1,...,d\}^k} \left( \int_{0 \leq t_1 \leq ... \leq t_k \leq t} \circ dB^{i_1}_{t_1} \cdots \circ dB^{i_k}_{t_k} \right) X_{i_1} \cdots X_{i_k}.$

Proof: Let us work on a fixed interval $[0,T]$ and consider a sequence $D_n$ of subdivisions of $[0,T]$ such that $D_{n+1} \subset D_n$ and whose mesh goes to 0 when $n \to +\infty$ .
As in the previous lecture, we denote by $B^n$ the piecewise linear process which is obtained from $B$ by interpolation along the subdivision $D_n$ , that is for $t_i^n \le t \le t_{i+1}^n$ ,
$B^n_t= \frac{t_{i+1}^n -t}{ t_{i+1}^n-t_i^n} B_{t_i} + \frac{t-t_i^n}{ t_{i+1}^n-t_i^n} B_{t_{i+1}}.$
We know from the previous lecture that $B^n$ converges to $B$ in the $p$ -rough paths topology $2 < p < 3$ . In particular all the iterated integrals $\int_{\Delta^k [s,t]} dB^{n,\otimes k}$ converge. We claim that actually,
$\lim_{n \to \infty} \int_{\Delta^k [s,t]} dB^{n,\otimes k}= \int_{\Delta^k[0,t]} \circ dB ^{\otimes k}.$
Let us denote
$\int_{\Delta^k[s,t]} \partial B ^{\otimes k}=\lim_{n \to \infty} \int_{\Delta^k [s,t]} dB^{n,\otimes k}.$
We are going to prove by induction on $k$ that $\int_{\Delta^k[s,t]} \partial B ^{\otimes k} =\int_{\Delta^k[s,t]} \circ dB ^{\otimes k}$ . We have
$\int_0^T B_s^n \otimes dB_s^{n} =\sum_{i=0}^{n-1} \int_{t_i^n}^{t_{i+1}^n} B_s^n \otimes dB_s^{n}$
$= \sum_{i=0}^{n-1} \int_{t_i^n}^{t_{i+1}^n} \left( \frac{t_{i+1}^n -s}{ t_{i+1}^n-t_i^n} B_{t^n_i} + \frac{s-t_i^n}{ t_{i+1}^n-t_i^n} B_{t^n_{i+1}}\right)ds \otimes \frac{ B_{t^n_{i+1}}- B_{t^n_{i}}}{ t_{i+1}^n-t_i^n}$
$= \frac{1}{2} \sum_{i=0}^{n-1} \left(B_{t^n_{i+1}}- B_{t^n_{i}} \right) \otimes \left( B_{t^n_{i+1}}+ B_{t^n_{i}}\right)$
By taking the limit when $t \to \infty$ , we deduce therefore that $\int_{\Delta^2[0,T]} \partial B ^{\otimes 2} =\int_{\Delta^2[0,T]} \circ dB ^{\otimes 2}$ . In the same way, we have for $0 \le s < t \le T$ , $\int_{\Delta^2[s,t]} \partial B ^{\otimes 2} =\int_{\Delta^2[s,t]} \circ dB ^{\otimes 2}$ . Assume now by induction, that for every $0 \le s \le t \le T$ and $1 \le j \le k$ , $\int_{\Delta^k[s,t]} \partial B ^{\otimes k} =\int_{\Delta^k[s,t]} \circ dB ^{\otimes k}$ . Let us denote
$\Gamma_{s,t}= \int_{\Delta^{k+1}[s,t]} \partial B ^{\otimes (k+1)} -\int_{\Delta^{k+1}[s,t]} \circ dB ^{\otimes (k+1)}.$
From the Chen’s relations, we immediately see that
$\Gamma_{s,u}=\Gamma_{s,t}+\Gamma_{t,u}.$
Moreover, it is easy to estimate
$\|\Gamma_{s,t} \| \le C \omega(s,t)^{\frac{k+1}{p}},$
where $2<p<3$ and $\omega (s,t)= \| \mathbf{B} \|_{p-var, [s,t]}$ , $\mathbf{B}$ being the lift of $B$ in the free Carnot group of step 2. Indeed, the bound
$\int_{\Delta^{k+1}[s,t]} \partial B ^{\otimes (k+1)} \le C_1 \omega(s,t)^{\frac{k+1}{p}},$
comes from the continuity of Lyons' lift and the bound
$\int_{\Delta^{k+1}[s,t]} \circ dB ^{\otimes (k+1)} \le C_2 \omega(s,t)^{\frac{k+1}{p}},$
easily comes from the Garsia-Rodemich-Rumsey inequality. As a conclusion, we deduce that $\Gamma_{s,t}=0$ which proves the induction $\square$

We finish this lecture by a very interesting probabilistic object, the expectation of the Brownian signature.
If
$Y=y_0+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k} a_{i_1,...,i_k} X_{i_1}...X_{i_k}.$
is a random series, that is if the coefficients are real random variables defined on a probability space, we will denote
$\mathbb{E}(Y)=\mathbb{E}(y_0)+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k} \mathbb{E}(a_{i_1,...,i_k}) X_{i_1}...X_{i_k}.$
as soon as the coefficients of $Y$ are integrable, where $\mathbb{E}$ stands for the expectation.

Theorem: For $t \ge 0$ ,
$\mathbb{E} \left( \mathfrak{S} (B)_t \right)=\exp \left( t \left(\frac{1}{2}\sum_{i=1}^d X_i^2 \right)\right).$

Proof:
An easy computation shows that if $\mathcal{I}_n$ is the set of words with length $n$ obtained by all the possible concatenations of the words $\{ (i,i) \}, \quad i \in \{1,...,d\}$ , then, if $I \notin \mathcal{I}_n$ then
$\mathbb{E} \left( \int_{\Delta^n [0,t]} \circ dB^I \right) =0$
and if $I \in \mathcal{I}_n$ then
$\mathbb{E} \left( \int_{\Delta^n [0,t]} \circ dB^I \right) =\frac{t^{\frac{n}{2}}}{2^{\frac{n}{2}}\left(\frac{n}{2} \right) ! }$
Therefore,
$\mathbb{E} \left( \mathfrak{S} (B)_t \right) =1+\sum_{k = 1}^{+\infty} \sum_{I \in \mathcal{I}_k} \frac{t^{\frac{k}{2}}}{2^{\frac{k}{2}}\left(\frac{k}{2} \right) ! } X_{i_1}...X_{i_k} = \exp \left( t\left(\frac{1}{2}\sum_{i=1}^d X_i^2 \right)\right)$
$\square$

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Lecture 27. Approximation of the Brownian rough path

Posted on March 28, 2013 by Fabrice Baudoin

Our goal in the next two lectures will be to prove that rough differential equations driven by a Brownian motion seen as a $p$ -rough path, $2 < p < 3$ are nothing else but stochastic differential equations understood in the Stratonovitch sense. The proof of this fact requires an explicit approximation of the Brownian rough path in the rough path topology which is interesting in itself.

Let $(B_t)_{t \ge 0}$ be a $n$ -dimensional Brownian motion and let us denote by
$\mathbf{B}_t=\left( B_t, \frac{1}{2} \left( \int_0^t B^i_sdB^j_s -B^j_sdB^i_s \right)_{1 \le i < j \le n} \right)$
its lift in the free Carnot group of step 2 over $\mathbb{R}^d$ .

Let us work on a fixed interval $[0,T]$ and consider a sequence $D_n$ of subdivisions of $[0,T]$ such that $D_{n+1} \subset D_n$ and whose mesh goes to 0 when $n \to +\infty$ . An example is given by the sequence of dyadic subdivisions. The family $\mathcal{F}_n =\sigma( B_t, t \in D_n )$ is then a filtration, that is an increasing family of $\sigma$ -fields. We denote by $B^n$ the piecewise linear process which is obtained from $B$ by interpolation along the subdivision $D_n$ , that is for $t_i^n \le t \le t_{i+1}^n$ ,
$B^n_t= \frac{t_{i+1}^n -t}{ t_{i+1}^n-t_i^n} B_{t_i} + \frac{t-t_i^n}{ t_{i+1}^n-t_i^n} B_{t_{i+1}}.$
The corresponding lifted process is then
$\mathbf{B}^n_t=\left( B^n_t, \frac{1}{2} \left( \int_0^t B^{n,i}_sdB^{n,j}_s -B^{n,j}_sdB^{n,i}_s \right)_{1 \le i < j \le n} \right) .$
The main result of the lecture is the following:

Theorem: Let $2 < p < 3$ . When $n \to +\infty$ , almost surely, $d_{p-var, [0,T]}( \mathbf{B}^n , \mathbf{B}) \to 0$ .

We split the proof in two lemmas.

Lemma: Let $t \in [0,T]$ . When $n \to +\infty$ , almost surely, $d( \mathbf{B}_t^n , \mathbf{B}_t) \to 0$ .

Proof: We first observe that, due to the Markov property of Brownian motion, we have for $t_i^n \le t \le t_{i+1}^n$ ,
$\mathbb{E} \left(B_t \mid \mathcal{F}_n\right)=\mathbb{E} \left(B_t \mid B_{t_i^n}, B_{t_i^{n+1}}\right) .$
It is then an easy exercise to check that
$\mathbb{E} \left(B_t \mid B_{t_i^n}, B_{t_i^{n+1}}\right) = \frac{t_{i+1}^n -t}{ t_{i+1}^n-t_i^n} B_{t_i} + \frac{t-t_i^n}{ t_{i+1}^n-t_i^n} B_{t_{i+1}}=B^n_t.$
As a conclusion, we get
$\mathbb{E} \left(B_t \mid \mathcal{F}_n\right)=B_t^n.$
It immediately follows that $B_t^n \to B_t$ when $n \to +\infty$ . In the same way, we have
$\mathbb{E} \left( \int_0^t B^i_sdB^j_s -B^j_sdB^i_s \mid \mathcal{F}_n\right)=\int_0^t B^{n,i}_sdB^{n,j}_s -B^{n,j}_sdB^{n,i}_s.$
Indeed, for $0 \le t < T$ and $\varepsilon$ small enough, we have by independence of $B^i$ and $B^j$ ,
$\mathbb{E} \left( B^i_t (B^j_{t+\varepsilon} -B^j_t) \mid \mathcal{F}_n\right)=\mathbb{E} \left( B^i_t\mid \mathcal{F}_n\right) \mathbb{E} \left( B^j_{t+\varepsilon} -B^j_t) \mid \mathcal{F}_n\right)=B^{n,i}_t (B^{n,j}_{t+\varepsilon} -B^{n,j}_t) ,$
and we conclude using the fact that Ito’s integral is a limit in $L^2$ of Riemann sums. It follows that, almost surely,
$\lim_{n \to \infty} \int_0^t B^{n,i}_sdB^{n,j}_s -B^{n,j}_sdB^{n,i}_s=\int_0^t B^i_sdB^j_s -B^j_sdB^i_s,$
and we conclude that almost surely, $d( \mathbf{B}_t^n , \mathbf{B}_t) \to 0$ $\square$

The second lemma is a uniform Holder estimate for $\mathbf{B}^n$ .

Lemma: For every $\alpha \in [0,1/2)$ , there exists a finite random variable $K$ that belongs to $L^p$ for every $p \ge 1$ and such that for every $0 \le s \le t \le T$ , and every $n \ge 1$ ,
$d(\mathbf{B}_s^n, \mathbf{B}_t^n) \le K | t-s|^{\alpha}.$

Proof: By using the theorem of equivalence of norms, we see that there is a constant $C$ such that
$d(\mathbf{B}_s^n, \mathbf{B}_t^n) \le C \left( \| B_t^n- B_s^n\| +\sum_{ i < j} \left| \int_s^t (B^{n,i}_u-B^{n,i}_s)dB^{n,j}_u -(B^{n,j}_u-B^{n,j}_s)dB^{n,i}_u\right|^{1/2} \right).$
From the Garsia-Rodemich-Rumsey inequality, we know that there is a finite random variable $K_1$ ( that belongs to $L^p$ for every $p \ge 1$ ), such that for every $0 \le s \le t \le T$ ,
$\left| \int_s^t (B^i_u-B^i_s)dB^j_u -(B^j_u-B^j_s)dB^i_u \right| \le K_1 | t-s|^{2\alpha}.$
Since
$\mathbb{E} \left( \int_s^t (B^i_u-B^i_s)dB^j_u -(B^j_u-B^j_s)dB^i_u \mid \mathcal{F}_n\right)=\int_s^t (B^{n,i}_u-B^{n,i}_s)dB^{n,j}_u -(B^{n,j}_u-B^{n,j}_s)dB^{n,i}_u,$
we deduce that
$\left|\int_s^t (B^{n,i}_u-B^{n,i}_s)dB^{n,j}_u -(B^{n,j}_u-B^{n,j}_s)dB^{n,i}_u \right| \le K_2 | t-s|^{2\alpha},$
where $K_2$ is a finite random variable that belongs to $L^p$ for every $p \ge 1$ . Similarly, of course, we have
$\| B_t^n- B_s^n\| \le K_3 | t-s|^{\alpha},$
and this completes the proof $\square$

We are now in position to finish the proof that, almost surely, $d_{p-var, [0,T]}( \mathbf{B}^n , \mathbf{B}) \to 0$ if $2 < p < 3$ . Indeed, if $t_i$ is a subdivision of $[0,T]$ , we have for $2 < p' < p$ ,
$\sum_{k=0}^{n-1} d\left( ( \mathbf{B}_{t_{i}}^n)^{-1} \mathbf{B}_{t_{i+1}}^n , ( \mathbf{B}_{t_{i}})^{-1} \mathbf{B}_{t_{i+1}}\right)^p \le d_{p'-var, [0,T]}( \mathbf{B}^n , \mathbf{B}) \left(\sup_{s,t} d\left( ( \mathbf{B}_{s}^n)^{-1} \mathbf{B}_{t}^n , ( \mathbf{B}_{s})^{-1} \mathbf{B}_{t}\right)\right)^{p-p'}$
By using the second lemma, it is seen that $d_{p'-var, [0,T]}( \mathbf{B}^n , \mathbf{B})$ is bounded when $n \to \infty$ and by combining the first two lemmas we easily see that $\sup_{s,t} d\left( ( \mathbf{B}_{s}^n)^{-1} \mathbf{B}_{t}^n , ( \mathbf{B}_{s})^{-1} \mathbf{B}_{t}\right) \to 0$ .

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Lecture 26. Lyons’ continuity theorem: Proof

Posted on March 25, 2013 by Fabrice Baudoin

We now turn to the proof of Lyons’ continuity theorem.

Theorem: Let $\gamma > p \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,x_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ such that
$\| S_{[p]}(x_1) \|^p_{p-var,[0,T]}+\| S_{[p]}(x_2) \|^p_{p-var,[0,T]} \le K$
with $K \ge 0$ .
Let $y_1,y_2$ be the solutions of the equations
$y_i(t)=y(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
There exists a constant $C$ depending only on $p,\gamma$ and $K$ such that for $0 \le s \le t \le T$ ,
$\|( y_2(t) -y_2(s)) - (y_1(t)-y_1(s))\| \le C \| V \|_{\text{Lip}^{\gamma}} e^{ C \| V \|^p_{\text{Lip}^{\gamma}}} d_{p-var,[0,T]} ( S_{[p]}(x_1), S_{[p]}(x_2)) \omega(s,t)^{1/p} ,$
where $\omega$ is the control
$\omega(s,t)=\left( \frac{d_{p-var,[s,t]} ( S_{[p]}(x_1), S_{[p]}(x_2)) }{d_{p-var; [0,T]}(S_{[p]}(x_1), S_{[p]}(x_2)) } \right)^p+\left( \frac{ \| S_{[p]}(x_1) \|_{p-var,[s,t]}}{\| S_{[p]}(x_1) \|_{p-var,[0,T]} } \right)^p+\left( \frac{ \| S_{[p]}(x_2) \|_{p-var,[s,t]}}{\| S_{[p]}(x_2) \|_{p-var,[0,T]} } \right)^p.$

Proof: We may assume $p < \gamma < [p]+1$ , and for conciseness of notations, we set $\varepsilon = d_{p-var,[0,T]} ( S_{[p]}(x_1), S_{[p]}(x_2))$ . Let
$g_i=\Delta_{\frac{1}{\omega(s,t)^{1/p}}} ( S_{[p]}(x_i)(s)^{-1} S_{[p]}(x_i)(t)), \quad i=1,2.$

We have,
$d(g_1,g_2) =\frac{1}{\omega(s,t)^{1/p}} d(S_{[p]}(x_1)(s)^{-1} S_{[p]}(x_1)(t) , S_{[p]}(x_2)(s)^{-1} S_{[p]}(x_2)(t))$
$\le \frac{1}{\omega(s,t)^{1/p}}d_{p-var,[s,t]} ( S_{[p]}(x_1), S_{[p]}(x_2))$
$\le \varepsilon$
and, in the same way,
$d(0,g_i) =\frac{1}{\omega(s,t)^{1/p}} d ( S_{[p]}(x_i)(s), S_{[p]}(x_i)(t))$
$=\frac{1}{\omega(s,t)^{1/p}} \| S_{[p]} (x_i) \|_{p-var,[s,t]} \le K.$

Therefore, there exist $x^{s,t}_1,x^{s,t}_2 \in C^{1-var}([s,t], \mathbb{R}^d)$ and a constant $C_1=C_1([p],K)$ such that
$S_{[p]}(x^{s,t}_i)(s)^{-1} S_{[p]}(x^{s,t}_i)(t) =S_{[p]}(x_i)(s)^{-1} S_{[p]}(x_i)(t) , i=1,2$
and
$\| x^{s,t}_1\|_{1-var,[s,t]} +\| x^{s,t}_2\|_{1-var,[s,t]} \le C_1\omega(s,t)^{1/p}$
and
$\| x^{s,t}_1-x^{s,t}_2 \|_{1-var,[s,t]} \le \varepsilon C_1\omega(s,t)^{1/p}.$
We define then $x_i^{s,t,u}$ as the concatenation of $x_i^{s,t}$ and $x_i^{t,u}$ . As in the proof of Davie’s lemma, we denote by $y_i^{s,t}$ the solution of the equation
$y^{s,t}_i(r)=y_i(s)+\sum_{j=1}^d \int_s^r V_j(y^{s,t}_i(v)) dx_i^j(v), \quad s \le r \le t, \quad i=1,2$
and consider the functionals
$\Gamma^i_{s,t}=(y_i(t)-y_i(s))-(y^{s,t}_i(t) -y^{s,t}_i(s))= y_i(t)-y^{s,t}_i(t),$
and
$\bar{\Gamma}_{s,t} =\Gamma^1_{s,t}-\Gamma^2_{s,t}$
From the proof of Davie’s estimate, it is seen that
$\| \Gamma^i_{s,t} \| \le \frac{1}{2} C_2 \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{[p]+1},$
and thus
$\| \bar{\Gamma}_{s,t} \| \le C_2 \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{[p]+1}.$
On the other hand, by estimating
$\bar{\Gamma}_{s,u}- \bar{\Gamma}_{s,t}- \bar{\Gamma}_{t,u},$
as in the proof of Davie’s lemma, that is by inserting $y_i^{s,t,u}$ which is the solution of the equation driven by the concatenation of $x_i^{s,t}$ and $x_i^{t,u}$ , and then by using the two lemmas of the previous lecture, we obtain the estimate
$\| \bar{\Gamma}_{s,u} \|$
$\le \| \bar{\Gamma}_{s,t} \| e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}} + \| \bar{\Gamma}_{t,u} \|+C_3( \| y_1-y_2\|_{\infty, [s,t]} +\varepsilon)\left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^\gamma e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}$
$\le \left( \| \bar{\Gamma}_{s,t} \| + \| \bar{\Gamma}_{t,u} \|+C_3( \| y_1-y_2\|_{\infty, [s,t]} +\varepsilon) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^{\gamma} \right) e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}.$
It remains to bound $\| y_1-y_2\|_{\infty, [s,t]}$ . For this let us observe that
$\| (y_1(t)-y_2(t))-(y_1(s)-y_2(s))-\bar{\Gamma}_{s,t} \| =\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\|.$
$\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\|$ can then be estimated by using classical estimates on differential equations driven by bounded variation paths. This gives,
$\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\| \le C_4 \left( \| y_1(s) -y_2(s) \| +\varepsilon \right) \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} e^{ C_4\| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p}}.$
By denoting $z=y_1-y_2$ , we can summarize the two above estimates as follows:
$\| \bar{\Gamma}_{s,u} \| \le \left( \| \bar{\Gamma}_{s,t} \| + \| \bar{\Gamma}_{t,u} \|+C_3( \| z\|_{\infty, [s,t]} +\varepsilon) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^{\gamma} \right) e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}$
and
$\| z(t)-z(s) -\bar{\Gamma}_{s,t} \| \le C_4 \left( \| z \|_{\infty,[0,s]} +\varepsilon \right) \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} e^{ C_4\| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p}}.$
From a lemma already used in the proof of Davie’s estimate, the first estimate implies
$\| \bar{\Gamma}_{s,t} \| \le C_5 \left( \varepsilon + \| z \|_{\infty,[0,t]} \right) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{\gamma} e^{ C_5 \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p}}.$
Using now the second estimate we obtain that for any interval $[a,b]$ included in $[0,T]$ ,
$\sup_{s,t \in [a,b]} \| z(t)-z(s) \| \le C_6 (\varepsilon + \| z \|_{\infty, [0,b]} ) \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p} e^{C_6 \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p}}.$
Using the fact that $z(0)=0$ and picking a subdivision $0 = \tau_0 \le \tau_1 \le \cdots \le \tau_N \le T$ such that
$C_6 \| V \|_{\text{Lip}^{\gamma}} e^{C_6 \| V \|_{\text{Lip}^{\gamma}} } \omega(\tau_i , \tau_{i+1} )^{1/p} \le 1/2$
we see that it implies
$\| z \|_{\infty,[0,T]} \le C_7 \varepsilon e^{C_7 \| V \|^p_{\text{Lip}^{\gamma}} }.$
Coming back to the estimate
$\sup_{s,t \in [a,b]} \| z(t)-z(s) \| \le C_6 (\varepsilon + \| z \|_{\infty, [0,b]} ) \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p} e^{C_6 \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p}}.$
concludes the proof $\square$

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Lecture 25. The Lyons’ continuity theorem: Preliminary lemmas

Posted on March 24, 2013 by Fabrice Baudoin

We now turn to the proof of the continuity theorem. We start with several lemmas, which are not difficult but a little technical. The first one is geometrically very intuitive.

Lemma: Let $g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ such that $d(g_1,g_2) \le \varepsilon$ with $\varepsilon > 0$ and $d(0,g_1), d(0,g_2) \le K$ with $K \ge 0$ . Then, there exists $x_1,x_2 \in C^{1-var}([0,1], \mathbb{R}^d)$ and a constant $C=C(N,K)$ such that $S_N(x)(1)=S_N(x)(0) g_i$ , $i=1,2$ and
$\| x_1\|_{1-var,[0,1]} +\| x_2\|_{1-var,[0,1]} \le C$
and
$\| x_1-x_2 \|_{1-var,[0,1]} \le \varepsilon C.$

Proof:
See the book by Friz-Victoir, page 161 $\square$

The next ingredient is the following estimate.

Lemma: Let $\gamma \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,\tilde{x}_1,x_2,\tilde{x}_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ such that
$S_{[\gamma]}(x_1)(T)=S_{[\gamma]}(\tilde{x}_1)(T), \quad S_{[\gamma]}(x_2)(T)=S_{[\gamma]}(\tilde{x}_2)(T).$

Let $y_1,y_2,\tilde{y}_1,\tilde{y}_2$ be the solutions of the equations
$y_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
and
$\tilde{y}_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(\tilde{y}_i(s)) d\tilde{x}_i^j(s), \quad 0 \le t \le T, \quad i=1,2.$
If
$\| x_1 \|_{1-var,[0,T]} +\| \tilde{x}_1 \|_{1-var,[0,T]} +\| x_2 \|_{1-var,[0,T]} +\| \tilde{x}_2 \|_{1-var,[0,T]} \le K$
and
$\| x_1-x_2 \|_{1-var, [0,T]}+ \| \tilde{x}_1-\tilde{x}_2 \|_{1-var, [0,T]} \le M,$
then, for some constant depending only on $\gamma$ ,
$\| (y_1(T)-\tilde{y}_1(T))-(y_2(T)-\tilde{y}_2(T)) \|$
$\le C \| y_1(0)-y_2(0) \| (\| V\|_{Lip^\gamma} K)^\gamma e^{C \| V\|_{Lip^\gamma} K}+CM \| V\|_{Lip^\gamma} (\| V\|_{Lip^\gamma} K)^\gamma e^{C \| V\|_{Lip^\gamma} K}$

Proof: Let us first observe that it is enough to prove the result when $\tilde{x}_1=\tilde{x_2}=0$ . Indeed, suppose that we can prove the result in that case. Define then the path $z$ to be the concatenation of $\tilde{x}_1(T-\cdot)$ and $x_1(\cdot)$ reparametrized so that $z:[0,T] \to \mathbb{R}^d$ . It is seen that the solution of the equation
$w(t)=\tilde{y}_1(T)+\sum_{j=1}^d \int_0^t V_j(w(s)) dz_i^j(s), \quad 0 \le t \le T$
satisfies
$w(T)-w(0)=y_1(T)-\tilde{y}_1(T).$
We thus assume that $\tilde{x}_1=\tilde{x_2}=0$ . In that case, from the assumption, we have
$S_{[\gamma]}(x_1)(T)=1, \quad S_{[\gamma]}(x_2)(T)=1.$
Taylor’s expansion gives then, with $n=[\gamma]$ ,
$y_1(T)-y_1(0)$
$=\int_{s \le r_1\le \cdots \le r_n \le t} \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y_1(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y_1(s)))dx^{i_1}_{1,r_1} \cdots dx^{i_n}_{1,r_n}.$
and similarly
$y_2(T)-y_2(0)$
$=\int_{s \le r_1\le \cdots \le r_n \le t} \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y_2(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y_2(s)))dx^{i_1}_{2,r_1} \cdots dx^{i_n}_{2,r_n}.$
The result is then easily obtained by using classical estimates for Riemann-Stieltjes integrals (details can be found page 230 in the book by Friz-Victoir) $\square$

Finally, the last lemma is an easy consequence of Gronwall’s lemma

Lemma: Let $\gamma \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,x_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ . Let $y_1,y_2,\tilde{y}_1,\tilde{y}_2$ be the solutions of the equations
$y_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
and
$\tilde{y}_i(t)=\tilde{y}_i(0)+\sum_{j=1}^d \int_0^t V_j(\tilde{y}_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2.$
If
$\| x_1 \|_{1-var,[0,T]} +\| x_2 \|_{1-var,[0,T]} \le K$
and
$\| x_1-x_2 \|_{1-var, [0,T]} \le M,$
then, for some constant depending only on $\gamma$ ,
$\| (y_1(T) -y_1(0)) - (\tilde{y}_1(T) -\tilde{y}_1(0)) - (y_2(T) -y_2(0)) + (\tilde{y}_2(T) -\tilde{y}_2(0)) \|$
$\le C \| V\|_{Lip^\gamma} K e^{ C \| V\|_{Lip^\gamma} K }\| y_1(0) - \tilde{y}_1(0) - y_2(0) + \tilde{y}_2(0) \| + C \| V\|_{Lip^\gamma} M e^{ C \| V\|_{Lip^\gamma} K }$
$+C \| V\|_{Lip^\gamma} K e^{ C \| V\|_{Lip^\gamma} K } ( \| y_1(0) - \tilde{y}_1(0)\| + \| y_2(0) - \tilde{y}_2(0) \| )^{\min (2,\gamma)-1} \left( \| \tilde{y}^1(0)-\tilde{y}^2(0)\|+ \| V\|_{Lip^\gamma} K \right)$

Proof: See the book by Friz-Victoir page 232 $\square$

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Lecture 24. The Lyons’ continuity theorem

Posted on March 21, 2013 by Fabrice Baudoin

We are now ready to state the main result of rough paths theory: the continuity of solutions of differential equations with respect to the driving path.

The proof will take us some time and will be preceeded by several lemmas. We can however already give the following important corollaries:

Corollary: [Lyon’s continuity theorem] Let $\gamma > p \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,x_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ such that
$\| S_{[p]}(x_1) \|^p_{p-var,[0,T]}+\| S_{[p]}(x_2) \|^p_{p-var,[0,T]} \le K$
with $K \ge 0$ .
Let $y_1,y_2$ be the solutions of the equations
$y_i(t)=y(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
There exists a constant $C$ depending only on $p,\gamma$ and $K$ such that for $0 \le s \le t \le T$ ,
$\| y_2-y_1\|_{p-var,[0,T]} \le C \| V \|_{\text{Lip}^{\gamma}} e^{ C \| V \|^p_{\text{Lip}^{\gamma}}} d_{p-var,[0,T]} ( S_{[p]}(x_1), S_{[p]}(x_2)) .$

This continuity statement immediately suggests the following basic definition for solutions of differential equation driven by $p$ -rough paths.

Theorem: Let $p \ge 1$ . Let $\mathbf{x}\in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ be a geometric $p$ -rough path over the $p$ -rough path $x$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ with $\gamma > p$ . If $\mathbf{x}_n \in C^{1-var} ([0,T],\mathbb{G}_{[p]} (\mathbb{R}^d))$ is a sequence that converges to $\mathbf{x}$ in $p$ -variation, then the solution of the equation

$y_n(t)=y(0)+\sum_{j=1}^d \int_0^t V_j(y_n(s)) dx_n^j(s), \quad 0 \le t \le T,$
converges in $p$ -variation to some $y \in C^{p-var} ([0,T], \mathbb{R}^d)$ that does not depend on the choice of the approximating sequence $\mathbf{x}_n$ and that we call a solution of the rough differential equation:
$y(t)=y(0)+\sum_{j=1}^d \int_0^t V_j(y(s)) dx^j(s), \quad 0 \le t \le T.$

The following propositions are easily obtained by a limiting argument:

Proposition:[Davie’s estimate for rough differential equations]
Let $\gamma > p \ge 1$ . Let $\mathbf{x}\in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ be a geometric $p$ -rough path over the $p$ -rough path $x$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $y$ be the solution of the rough differential equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$ ,
$\| y \|_{p-var, [s,t]} \le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \| \mathbf{x} \|_{p-var,[s,t]} +\| V \|^p_{\text{Lip}^{\gamma-1}} \| \mathbf{x} \|^p_{p-var,[s,t]} \right).$

Proposition: Let $\gamma > p \ge 1$ . Let $\mathbf{x}\in \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ be a geometric $p$ -rough path over the $p$ -rough path $x$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $y$ be the solution of the rough differential equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s \le t \le T$ ,
$\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|$
$\le C \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| \mathbf{x} \|^\gamma_{p-var,[s,t]}$

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Lecture 23. Davie’s estimate (2)

Posted on March 17, 2013 by Fabrice Baudoin

We now turn to the proof of Davie’s estimate. We follow the approach by Friz-Victoir who smartly use interpolations by geodesics in Carnot groups.

Theorem: Let $\gamma > p \ge 1$ . Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x \in C^{1-var}([0,T], \mathbb{R}^d)$ . Let $y$ be the solution of the equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$ ,
$\| y \|_{p-var, [s,t]} \le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]} +\| V \|^p_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^p_{p-var,[s,t]} \right),$
where $S_{[p]} (x)$ is the lift of $x$ in $\mathbb{G}_{[p]}(\mathbb{R}^d)$ .

Proof: For $s \le t$ , we denote by $x^{s,t}$ a path in $C^{1-var}([s,t],\mathbb{R}^d)$ such that $S_{[\gamma]}( x^{s,t})(s)=S_{[\gamma]}( x)(s)$ , $S_{[\gamma]}( x^{s,t})(t)=S_{[\gamma]}( x )(t)$ and $S_{[\gamma]}( x^{s,t})(u)$ , $s\le u \le t$ , is a geodesic for the Carnot-Caratheodory distance. We consider then $y^{s,t}$ to be the solution of the equation
$y^{s,t} (u)=y(s)+\sum_{i=1}^d \int_s^u V_i(y^{s,t} (v)) dx^i(v), \quad s \le u \le t.$
We can readily observe that from the continuity of Lyons’ lift:
$\| x^{s,t} \|_{1-var, [s,t]} = d( S_{[\gamma]}( x)(s), S_{[\gamma]}( x)(t)) \le \| S_{[\gamma]} (x) \|_{p-var,[s,t]} \le K \| S_{[p]} (x) \|_{p-var,[s,t]} .$
Let us now denote
$\Gamma_{s,t} =(y(t) -y(s)) -( y^{s,t} (t) -y^{s,t} (s)).$
For fixed $s \le t \le u$ , we have then:
$\Gamma_{s,u}-\Gamma_{s,t}-\Gamma_{t,u}=(y^{s,u}(s)-y^{s,u}(u))-( y^{s,t}(s)-y^{s,t}(t))-(y^{t,u}(t)-y^{t,u}(u)).$
To estimate this quantity, we consider the path $y^{s,t,u}(v)$ , $s \le v \le u$ , that solves the ordinary differential equation driven by the concatenation of $x^{s,t}$ and $x^{t,u}$ . We first estimate $y^{s,t,u}(u)-y^{s,u}(u)$ by observing that $y^{s,t,u}(u)$ and $y^{s,u}(u)$ have the same Taylor expansion up to order $[\gamma]$ . Thus by using the lemma of the previous lecture and the triangle inequality, we easily get that:
$\| y^{s,t,u}(u)-y^{s,u}(u) \| \le C_1 \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \left( \int_s^t \| dx^{s,t} (r)\| + \int_t^u \| dx^{t,u} (r)\| \right)^\gamma$
$\le C_2 \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^\gamma_{p-var,[s,u]} .$
We then estimate $(y^{s,t,u}(u) - y^{s,t,u}(s)) + ( y^{s,t}(s)-y^{s,t}(t))+(y^{t,u}(t)-y^{t,u}(u))$ by observing that $y^{s,t,u}(s)=y^{s,t}(s)$ , $y^{s,t,u}(t)=y^{s,t}(t)$ . Thus,
$(y^{s,t,u}(u) - y^{s,t,u}(s)) + ( y^{s,t}(s)-y^{s,t}(t))+(y^{t,u}(t)-y^{t,u}(u))$
$= (y^{s,t,u} (u)- y^{s,t,u} (t))-( y^{t,u}(u) - y^{t,u}(t) )$
This last term is estimated by using basic continuity estimates with respect to the initial condition which gives
$\| (y^{s,t,u} (u)- y^{s,t,u} (t))-( y^{t,u}(u) - y^{t,u}(t) )\|$
$\le \|y^{s,t,u} (t) -y^{t,u}(t) \| \| V \|_{\text{Lip}^{\gamma-1}} \int_t^u \| dx^{t,u}(r) \| \exp \left( \| V \|_{\text{Lip}^{\gamma-1}} \int_t^u \| dx^{t,u}(r)\|\right)$
$\le C_3 \| \Gamma_{s,t} \| \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[t,u]} \exp \left( C_3 \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[t,u]} \right)$
We conclude
$\| \Gamma_{s,u}-\Gamma_{s,t}-\Gamma_{t,u}\| \le C_2 \omega(s,u)^{\gamma/p} +C_3 \| \Gamma_{s,t} \| \omega(t,u)^{1/p} \exp \left(C_3 \omega(t,u)^{1/p} \right),$
where
$\omega(s,t)= \left( \| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]}\right)^p.$
The basic inequality $1+x e^x \le e^{2x}$ combined with the triangle inequality gives:
$\| \Gamma_{s,u} \| \le \| \Gamma_{t,u} \|+\|\Gamma_{s,t}\| \exp\left(2C_3 \omega(s,u)^{1/p} \right)+ C_2 \omega(s,u)^{\gamma/p}$
$\le \left( \| \Gamma_{t,u} \|+\|\Gamma_{s,t}\| + C_2 \omega(s,u)^{\gamma/p} \right) \exp\left(2C_3 \omega(s,u)^{1/p} \right)$ .

We are now in position to apply the lemma of the previous lecture (we let the reader check that the assumptions are satisfied). We deduce then
$\| \Gamma_{s,t} \| \le C_4 \omega(s,t)^{\gamma /p} \exp\left(C_4 \omega(s,t)^{1 /p} \right).$
We now keep in mind that
$\Gamma_{s,t} =(y(t) -y(s)) -( y^{s,t} (t) -y^{s,t} (s)),$
and $y^{s,t} (t) -y^{s,t} (s)$ can be estimated by using basic estimates on differential equations:
$\| y^{s,t} (t) -y^{s,t} (s) \| \le C_5 \| V \|_{\text{Lip}^{\gamma-1}} \int_s^t \| dx^{s,t}(u) \|$
$\le C_6 \omega(s,t)^{1/p}.$
From the triangle inequality, we conclude then:
$\| y(s)-y(t) \| \le C_6 \omega(s,t)^{1/p}+ C_4 \omega(s,t)^{\gamma /p} \exp\left(C_4 \omega(s,t)^{1 /p} \right),$
In particular we have for $s,t$ such that $\omega(s,t)\le 1$ ,
$\| y(s)-y(t) \| \le C_7 \omega(s,t)^{1/p}.$
This easily gives the required estimate (see Proposition 5.10 in the book by Friz-Victoir) $\square$

We can remark that the proof actually also provided the following estimate which is interesting in itself:

Proposition: Let $\gamma > p \ge 1$ . Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x \in C^{1-var}([0,T], \mathbb{R}^d)$ . Let $y$ be the solution of the equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$ ,
$\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|$
$\le C \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^\gamma_{p-var,[s,t]}$ .

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Lecture 22. Davie’s estimate (1)

Posted on March 17, 2013 by Fabrice Baudoin

In this Lecture, we prove one of the fundamental estimates of rough paths theory. This estimate is due to Davie. It provides a basic estimate for the solution of the differential equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s)$
in terms of the $p$ -variation of the lift of $x$ in the free Carnot group of step $[p]$ .

We first introduce the somehow minimal regularity requirement on the vector fields $V_i$ ‘s to study rough differential equations.

Definition. A vector field $V$ on $\mathbb{R}^n$ is called $\gamma$ -Lipschitz if it is $[\gamma]$ times continuously differentiable and there exists a constant $M \ge 0$ such that the supremum norm of its $k$ th derivatives $k=0, \cdots, [\gamma]$ and the $\gamma-[\gamma]$ Holder norm of its $[\gamma]$ th derivative are bounded by $M$ . The smallest $M$ that satisfies the above condition is the $\gamma$ -Lipschitz norm of $V$ and will be denoted $\| V \|_{\text{Lip}^\gamma}$ .

The fundamental estimate by Davie is the following;

Definition: Let $\gamma > p \ge 1$ . Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x \in C^{1-var}([0,T], \mathbb{R}^d)$ . Let $y$ be the solution of the equation
$y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.$
There exists a constant $C$ depending only on $p$ and $\gamma$ such that for every $0 \le s < t \le T$ ,
$\| y \|_{p-var, [s,t]} \le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]} +\| V \|^p_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^p_{p-var,[s,t]} \right),$
where $S_{[p]} (x)$ is the lift of $x$ in $\mathbb{G}_{[p]}(\mathbb{R}^d)$ .

We start with two preliminary lemmas, the first one being interesting in itself.

Lemma: Let $\gamma > 1$ . Assume that $V_1, \cdots, V_d$ are $(\gamma-1)$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x \in C^{1-var}([s,t], \mathbb{R}^d)$ . Let $y$ be the solution of the equation
$y(v)=y(s)+\sum_{i=1}^d \int_s^v V_i(y(u)) dx^i(u), \quad s \le v \le t.$
There exists a constant $C$ depending only on $\gamma$ such that,
$\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|$
$\le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \int_s^t \| dx_r\| \right)^\gamma,$
where $\mathbf{I}$ is the identity map.

Proof: For notational simplicity, we denote $n=[\gamma]$ . An iterative use of the change of variable formula leads to
$y(t)-y(s)-\sum_{k=1}^{n} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k}$
$=\int_{s < r_1 < \cdots < r_n < t} \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s)))dx^{i_1}_{r_1} \cdots dx^{i_n}_{r_n}.$
Since $V_1, \cdots, V_d$ are $(\gamma-1)$ -Lipschitz, we deduce that
$\| V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s))\| \le \| V \|_{\text{Lip}^{\gamma-1}}^n \|y(r_1)-y(s)\|^{\gamma-n}.$
Since,
$\|y(r_1)-y(s)\|\le \| V \|_{\text{Lip}^{\gamma-1}} \int_s^{r_1} \| dx_r\|,$
we deduce that
$\| V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s))\| \le \| V \|_{\text{Lip}^{\gamma-1}}^\gamma \left( \int_s^{t} \| dx_r\| \right)^{\gamma-n}.$
The result follows then easily by plugging this estimate into the integral
$\int_{s < r_1 < \cdots < r_n < t} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s)))dx^{i_1}_{r_1} \cdots dx^{i_n}_{r_n}$ $\square$

The second lemma is an analogue of a result already used in previous lectures (Young-Loeve estimate, estimates on iterated integrals).

Lemma: Let $\Gamma: \{ 0 \le s \le t \le T \} \to \mathbb{R}^n$ . Let us assume that:

There exists a control $\tilde{\omega}$ such that
$\lim_{r \to 0} \sup_{(s,t)\in \Gamma, \tilde{\omega}(s,t) \le r } \frac{\| \Gamma_{s,t} \|}{r}=0;$

There exists a control $\omega$ and $\theta > 1, \xi > 0, K \ge 0, \alpha > 0$ such that for $0 \le s \le t \le u\le T$ ,
$\| \Gamma_{s,u} \| \le \left( \| \Gamma_{s,t} \|+ \| \Gamma_{t,u} \| +\xi \omega(s,u)^\theta\right)\exp( K \omega(s,t)^\alpha).$

Then, for all $0 \le s < t \le T$ ,
$\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega(s,t)^\theta \exp\left( \frac{2K}{1-2^{-\alpha}} \omega(s,u)^\alpha\right).$

Proof:
For $\varepsilon > 0$ , consider then the control
$\omega_\varepsilon (s,t)= \omega(s,t) +\varepsilon \tilde{\omega}(s,t)$
Define now
$\Psi(r)= \sup_{s,u, \omega_\varepsilon (s,u)\le r} \| \Gamma_{s,u}\|.$
If $s,u$ is such that $\omega_\varepsilon (s,u) \le r$ , we can find a $t$ such that $\omega_\varepsilon(s,t) \le \frac{1}{2} \omega_\varepsilon(s,u)$ , $\omega_\varepsilon(t,u) \le \frac{1}{2} \omega_\varepsilon(s,u)$ . Indeed, the continuity of $\omega_\varepsilon$ forces the existence of a $t$ such that $\omega_\varepsilon(s,t)=\omega_\varepsilon(t,u)$ . We obtain therefore
$\| \Gamma_{s,u}\|\le \left( 2 \Psi(r/2) + \xi r^\theta \right) \exp (K r^\alpha) ,$
which implies by maximization,
$\Psi(r)\le \left( 2 \Psi(r/2) + \xi r^\theta \right) \exp (K r^\alpha).$
We have $\lim_{r \to 0} \frac{\Psi (r)}{r} =0$ and an iteration easily gives
$\Psi (r) \le \frac{\xi}{1-2^{1-\theta}}r^\theta \exp\left( \frac{2K}{1-2^{-\alpha}} r^\alpha\right).$
We deduce
$\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega_\varepsilon (s,t)^\theta \exp\left( \frac{2K}{1-2^{-\alpha}} \omega_\varepsilon (s,u)^\alpha\right),$
and the result follows by letting $\varepsilon \to 0$ $\square$

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What is a mathematician ?

Posted on February 12, 2013 by Fabrice Baudoin

For those who understand French, here is an inspired explanation right to the point by Alain Connes about the work of mathematicians

The two following videos by André Lichnerowicz also contain thoughts about the nature of mathematics. I had the chance to meet Pr. Lichnerowicz when I was about 15 since he was a friend of my grandmother. I was already curious about mathematics at that age and he took a lot of time to discuss with me about the philosophy of mathematics and thus played a decisive role in my vocation.

A. Connes and A. Lichnerowicz confront their different opinions about the nature of mathematics in the beautiful book Triangle de Pensées.

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Lecture 21. The Brownian motion as a rough path (2)

Posted on January 31, 2013 by Fabrice Baudoin

In the previous Lecture we proved that Brownian motion paths almost surely have a bounded $p$ -variation for every $p > 2$ . In this lecture, we are going to prove that they even almost surely are $p$ -rough paths for $2 < p < 3$ . To prove this, we need to construct a geometric $p$ rough path over the Brownian motion, that is we need to lift the Brownian motion to the free nilpotent Lie group of step $2$ , $\mathbb{G}_{2} (\mathbb{R}^d)$ . In this process, we will have to define the iterated integrals $\int dB^{\otimes 2}=\int B \otimes dB$ . This can be done by using the theory of stochastic integrals. Indeed, it is well known (and easy to prove !) that if
$\Delta_n [0,t]=\left\{ 0=t^n_0 \le t^n_1 \le ...\le t^n_n=t \right\}$
is a subdivision of the time interval $[0,t]$ whose mesh goes to $0$ , then the Riemann sums
$\sum_{k=0}^{n-1} B_{t_k^n} \otimes (B_{t_{k+1}^n}-B_{t_k^n})$
converge in probability to a random variable denoted $\int_0^t B_s \otimes dB_s$ . We can then prove that the stochastic process $\int_0^t B_s \otimes dB_s$ admits a continuous version which is a martingale. With this integral of $B$ against itself in hands, we can now proceed to construct the canonical geometric rough path over $B$ .

Let $d \geq 2$ and denote $\mathcal{AS}_d$ the space of $d \times d$ skew-symmetric matrices. We can realize the group $\mathbb{G}_{2} (\mathbb{R}^d )$ in the following way
$\mathbb{G}_{2} (\mathbb{R}^d ) = ( \mathbb{R}^d \times \mathcal{AS}_d ,\circledast)$
where $\circledast$ is the group law defined by
$( \alpha_1 , \omega_1 ) \circledast ( \alpha_2 , \omega_2 )= ( \alpha_1 + \alpha_2 , \omega_1 + \omega_2 + \frac{1}{2} \alpha_1 \wedge \alpha_2 ).$
Here we use the following notation; if $\alpha_1, \alpha_2 \in \mathbb{R}^d$ , then $\alpha_1 \wedge \alpha_2$ denotes the skew-symmetric matrix $\left( \alpha_1^i \alpha_2^j - \alpha_1^j \alpha_2^i \right)_{i,j}$ . Notice that the dilation writes
$\label{scaling 2 step} c \cdot ( \alpha , \omega ) = ( c \alpha , c^2 \omega ).$

Remark: If $x:[0,+\infty) \rightarrow \mathbb{R}^2$ is a continuous path with bounded variation then for $0 < t_1 < t_2$ we denote
$\Delta_{[t_1,t_2]}x=\left( x^1_{t_2}-x^1_{t_1},x^2_{t_2}-x^2_{t_1},S_{[t_1,t_2]}x \right),$
where $S_{[t_1,t_2]}x$ is the area swept out by the vector $\overrightarrow{x_{t_1}x_t}$ during the time interval $[t_1,t_2]$ . Then, it is easily checked that for $0 < t_1 < t_2 < t_3$ ,
$\Delta_{[t_1,t_3]}x=\Delta_{[t_1,t_2]}x \circledast \Delta_{[t_2,t_3]}x,$
where $\circledast$ is precisely the law of $\mathbb{G}_{2} (\mathbb{R}^2 )$ , i.e. for $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2) \in \mathbb{R}^3$ ,
$(x_1,y_1,z_1) \circledast (x_2,y_2,z_2)=\left( x_1+x_2,y_1+y_2,z_1+z_2+\frac{1}{2} \left(x_1 y_2 - x_2 y_1 \right) \right).$

We now are in position to give the fundamental definition.
Definition: The process
$\mathbf{B}_{t}=\left( B_t , \frac{1}{2} \left( \int_0^t B^i_s dB^j_s-B^j_s dB^i_s \right)_{1 \leq i,j \leq d} \right), \text{ }t \geq 0.$
is called the lift of the Brownian motion $(B_{t})_{ t \geq 0}$ in the group $\mathbb{G}_{2}(\mathbb{R}^d )$ .

Interestingly, it turns out that the lift of a Brownian motion is a Markov process. Indeed, consider the vector fields
$D_i (x)=\frac{\partial}{\partial x^i}+ \frac{1}{2} \sum_{j < i} x^j \frac{\partial}{\partial x^{j,i}}- \frac{1}{2} \sum_{j > i} x^j \frac{\partial}{\partial x^{i,j}}, \text{ }1 \leq i \leq d,$
defined on $\mathbb{R}^d \times \mathcal{AS}_d$ . It is easy to check that:

For $x \in \mathbb{R}^d \times \mathcal{AS}_d$ ,
$[ D_i , D_j ](x)= \frac{\partial}{\partial x^{i,j}}, \text{ } 1 \leq i < j \leq d;$
For $x \in \mathbb{R}^d \times \mathcal{AS}_d$ ,
$[[ D_i , D_j],D_k ](x)= 0, \text{ }1 \leq i ,j,k \leq d;$
The vector fields $\left( D_i , [ D_j , D_k ] \right)_{1 \leq i \leq d, 1 \leq j < k \leq d}$
are invariant with respect to the left action of $\mathbb{G}_{2} (\mathbb{R}^d )$ on itself and form a basis of the Lie algebra $\mathfrak{g}_{2} (\mathbb{R}^d)$ of $\mathbb{G}_{2} (\mathbb{R}^d )$ .

The process $(\mathbf{B}_{t})_{ t \geq 0}$ solves the Stratonovitch stochastic differential equation
$d\mathbf{B}_t=\sum_{i=1}^d D_i (\mathbf{B}_t) \circ dB^i_s.$
and as such, is a diffusion process in $\mathbb{R}^d \times \mathcal{AS}_d$ whose generator is the subelliptic diffusion operator given by $\sum_{i=1}^d D_i^2$ .

Finally, also observe that we have the following scaling property, for every $c> 0$,
$\left( \mathbf{B}_{ct} \right)_{t \geq 0} =^{\text{law}} \left(\sqrt{c} \cdot \mathbf{B}_{t} \right)_{t \geq 0}.$

Before we turn to the fundamental result of this Lecture, we need the following result which is known as the Garsia-Rodemich-Rumsey inequality (see the proof page 573 in the book by Friz-Victoir):

Lemma: Let $(X,d)$ be a metric space and $x:[0,T] \to E$ be a continuous path. Let $q > 1$ and $\alpha \in (1/q,1)$ . There exists a constant $C=C(\alpha,q)$ such that:
$d(x(s),x(t))^q\le C |t-s|^{\alpha q -1} \int_{[s,t]^2} \frac{ d(x(u),x(v))^q}{|u-v|^{1+\alpha q}} du dv.$

Theorem: The paths of $(\mathbf{B}_{t})_{ t \geq 0}$ are almost surely geometric $p$ -rough paths for $2 < p < 3$ . As a consequence, the Brownian motion paths almost surely are $p$ -rough paths for $2 < p < 3$ . Let $q > 1$ .

Proof: We know that if $q < p$ , then $C_0^{q-var} ([0,T], \mathbb{G}_{[p]} (\mathbb{R}^d)) \subset \mathbf{\Omega G}^p([0,T],\mathbb{R}^d)$ . Therefore, we need to prove that for $2 < p < 3$ , the paths of $(\mathbf{B}_{t})_{ t \geq 0}$ almost surely have bounded $p$ -variation with respect to the Carnot-Caratheodory distance. From the scaling property of $(\mathbf{B}_{t})_{ t \geq 0}$ and of the Carnot-Caratheodory distance, we have in distribution
$d( \mathbf{B}_{s},\mathbf{B}_{t})=^d \sqrt{ t-s} d (0, \mathbf{B}_1).$
Moreover, from the equivalence of homogeneous norms, we have
$d (0, \mathbf{B}_1) \simeq \| B_1\| + \left\| \int_0^1 B \otimes dB \right\|^{1/2}.$
It easily follows from that, that for every $q > 1$ ,
$\mathbb{E} \left( \frac{ d( \mathbf{B}_{s},\mathbf{B}_{t})^q }{ (t-s)^{q/2} } \right)=\mathbb{E} \left( d (0, \mathbf{B}_1)^q\right) < +\infty.$
Thus, from Fubini’s theorem we obtain
$\mathbb{E} \left( \int_{[0,T]^2} \frac{ d(\mathbf{B}_{u},\mathbf{B}_{v})^q}{|u-v|^{q/2}} du dv \right)< +\infty.$
The Garsia-Rodemich-Rumsey inequality implies then
$d( \mathbf{B}_{s},\mathbf{B}_{t})^q\le C |t-s|^{q/2 -1} \int_{[0,T]^2} \frac{ d(\mathbf{B}_{u},\mathbf{B}_{v})^q}{|u-v|^{q/2}} du dv.$
Therefore, the paths of $(\mathbf{B}_{t})_{ t \geq 0}$ almost surely have bounded $p$ -variation for $p > 2$ $\square$