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]}

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