Lecture 25. The Lyons’ continuity theorem: Preliminary lemmas

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

This entry was posted in Rough paths theory. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s