Lecture 20. The Brownian motion as a rough path (1)

It is now time to give a fundamental example of rough path: The Brownian motion. As we are going to see, a Brownian motion is a p-rough path for any 2 < p < 3.

We first remind the following basic definition.
Definition: Let (\Omega,\mathcal{F},\mathbb{P}) be a probability space. A continuous d-dimensional process (B_t)_{t \ge 0} is called a standard Brownian motion if it is a Gaussian process with mean function
and covariance function
R(s,t)=\mathbb{E}(B_s \otimes B_t)=\min (s,t) \mathbf{I}_d.

For a Brownian motion (B_t)_{t \ge 0}, the following properties are easy to check:

  • B_0=0 a.s.;
  • For any h \geq 0, the process (B_{t+h} - B_h)_{t \ge 0} is a standard Brownian motion;
  • For any t > s\geq 0, the random variable B_t -B_s is independent of the \sigma-algebra \sigma(B_u, u \le s ).
  • For every c > 0, the process (B_{ct})_{t \geq 0} has the same law as the process (\sqrt{c} B_t)_{t \geq 0}.

An easy computation shows that for n \ge 0 and 0 \le s \le t:
\mathbb{E} \left( \|B_t - B_s\|^{2n} \right)=\frac{(2n)!}{2^n n!} (t-s)^n.
Therefore, as a consequence of the Kolmogorov continuity theorem, for any T \ge 0 and 0 \le  \varepsilon < 1/2, there exists a finite random variable C_{T, \varepsilon} such that for 0 \le s \le t \le T,
\| B_t -B_s \| \le C_{T, \varepsilon} |t-s|^{1/2-\varepsilon }.
We deduce in particular that for any p > 2, we have almost surely
\| B\|_{p-var,[0,T]} < +\infty.

We now prove that for 1 \le p < 2, we have almost surely
\| B\|_{p-var,[0,T]}=+\infty.

In the sequel, 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], we denote by
\mid\Delta_n [0,t] \mid=\max \{ \mid t^n_{k+1}-t^n_k \mid , k=0,...,n-1 \},
the mesh of this subdivision.

Proposition: Let (B_t)_{t\ge 0} be a standard Brownian motion. Let t \ge 0. For every sequence \Delta_n [0,t] of subdivisions such that
\lim_{n \rightarrow +\infty}\mid\Delta_n [0,t]\mid=0,
the following convergence takes place in L^2 (and thus in probability),
\lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \left\| B_{t^n_k}-B_{t^n_{k-1}}\right\|^2=t.
As a consequence, if 1 \le p < 2, for every T \ge 0, almost surely,
\| B\|_{p-var,[0,T]}=+\infty.

Proof: We prove the result in dimension 1 and let the reader adapt it to the multidimensional setting.
Let us denote
V_n=\sum_{k=1}^{n} \left( B_{t^n_k} -B_{t^n_{k-1}}\right)^2.
Thanks to the stationarity and the independence of Brownian increments, we have:
\mathbb{E} \left( (V_n-t)^2\right)=\mathbb{E} \left( V_n^2\right)-2t\mathbb{E} \left( V_n\right)+t^2
=\sum_{j,k=1}^n\mathbb{E} \left( \left( B_{t^n_j} -B_{t^n_{j-1}}\right)^2\left( B_{t^n_k}   -B_{t^n_{k-1}}\right)^2\right)-t^2
=\sum_{k=1}^n\mathbb{E} \left( \left( B_{t^n_j} -B_{t^n_{j-1}}\right)^4\right)+2\sum_{1\le j<k\le n}^n\mathbb{E} \left( \left( B_{t^n_j} -B_{t^n_{j-1}}\right)^2\left( B_{t^n_k} -B_{t^n_{k-1}}\right)^2\right)-t^2
=\sum_{k=1}^n (t^n_k-t^n_{k-1})^2 \mathbb{E} \left( B_1^4\right)+2\sum_{1\le j<k\le n}^n (t^n_j-t^n_{j-1})(t^n_k-t^n_{k-1})-t^2
=3\sum_{k=1}^n (t^n_j-t^n_{j-1})^2+2\sum_{1\le j<k\le n}^n (t^n_j-t^n_{j-1})(t^n_k-t^n_{k-1})-t^2
=2\sum_{k=1}^n (t^n_k-t^n_{k-1})^2
\le 2t\mid\Delta_n [0,t]\mid \rightarrow_{n \rightarrow +\infty} 0.

Let us now prove that, as a consequence of this convergence, the paths of the process (B_t)_{t\ge 0} almost surely have an infinite p-variation on the time interval [0,t] if 1 \le p < 2. Reasoning by absurd, let us assume that \| B \|_{p-var,[0,t]} \le M . From the above result, since the convergence in probability implies the existence of an almost surely convergent subsequence, we can find a sequence of subdivisions \Delta_n  [0,t] whose mesh tends to 0 and such that almost surely,
\lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \left( B_{t^n_k}-B_{t^n_{k-1}}\right)^2=t.
We get then
\sum_{k=1}^{n} \left( B_{t^n_k} -B_{t^n_{k-1}}\right)^2 \le M^p \sup_{1\le k \le n} \mid B_{t^n_k} -B_{t^n_{k-1}} \mid^{2-p} \rightarrow_{n \rightarrow +\infty} 0,
which is clearly absurd \square

Therefore only the case p=2 is let open. It is actually possible to prove that:

Proposition: For every T \ge 0, we have almost surely
\| B \|_{2-var,[0,T]} = +\infty.

Proof: See the book by Friz-Victoir page 381 \square

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