## Lecture 24. The Lyons’ continuity theorem

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.

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
$\omega(s,t)=\left( \frac{d_{p-var,[s,t]} ( S_{[p]}(x_1), S_{[p]}(x_2)) }{d_{p-var; [0,T]}(S_{[p]}(x_1), S_{[p]}(x_2)) } \right)^p+\left( \frac{ \| S_{[p]}(x_1) \|_{p-var,[s,t]}}{\| S_{[p]}(x_1) \|_{p-var,[0,T]} } \right)^p+\left( \frac{ \| S_{[p]}(x_2) \|_{p-var,[s,t]}}{\| S_{[p]}(x_2) \|_{p-var,[0,T]} } \right)^p.$

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 $0 \le s \le t \le T$,
$\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|$
$\le C \| V \|^\gamma_{\text{Lip}^{\gamma-1}} \| \mathbf{x} \|^\gamma_{p-var,[s,t]}$

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