Lecture 1. An overview of rough paths theory

Let us consider a differential equation that writes
y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),
where the V_i‘s are vector fields on \mathbb{R}^n and where the driving signal x(t)=(x^1(t), \cdots, x^d(t)) is a continuous bounded variation path. If the vector fields are Lipschitz continuous then, for any fixed initial condition, there is a unique solution y(t) to the equation. We can see this solution y as a function of the driving signal x. It is an important question to understand for which topology, this function is continuous.

A simple example shows that the topology of uniform convergence is not the correct one here. Indeed, let us consider the differential equation
y_1(t) = x_1(t)
y_2(t)= x_2(t)
y_3(t)= -\int_0^t y_2(s) dx_1(s) +\int_0^t y_1(s) dx_2(s)
where
x_1(t)=\frac{1}{n} \cos (n^2 t ), \quad x_2(t)=\frac{1}{n} \sin (n^2 t).
A straightforward computation shows that y_3(t)=t. When n \to \infty, (x_1,x_2) converges uniformly to 0 whereas, of course, (y_1,y_2,y_3) does not converge to 0. In this framework , a correct topology is given by the topology of convergence in 1-variation on compact sets. To fix the ideas, let us work on the interval [0,1]. The distance in 1-variation between two continuous bounded variation paths x,\tilde{x}:[0,1] \to \mathbb{R}^d is given by
\delta_1(x,\tilde{x})=\|x(0)-\tilde{x}(0) \|+ \sup_{\pi}  \sum_{k=0}^{n-1} \| (x(t_{i+1})-\tilde{x}(t_{i+1})) -(x(t_i)-\tilde{x}(t_i))  \|,
where the supremum is taken over all the subdivisions \pi =\{ 0 \leq t_1 \leq \cdots \leq t_n \leq 1 \}.

It is then a fact that is going to be proved in this class that if the V_i‘s are bounded and if x^n:[0,1] \to \mathbb{R}^d is a sequence of bounded variation paths that converges in 1-variation to a continuous path x with bounded variation, then the solutions of the differential equations
y^n(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y^n(s)) dx^{i,n}(s),
converge in 1-variation to the solution of
y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).
This type of continuity result suggests to use a topology in p-variation, p \ge 1, to try to extend the map x \to y to a larger class of driving signals x. More precisely, for p \geq 1, let us denote by \Omega^p  (\mathbb{R}^d) the closure of the set of continuous with bounded variation paths x:[0,1] \rightarrow  \mathbb{R}^d with respect to the distance in p-variation which is given by
\delta_p(x,\tilde{x})=\left( \|x(0)-\tilde{x}(0) \|^p+ \sup_{\pi}  \sum_{k=0}^{n-1} \| (x(t_{i+1})-\tilde{x}(t_{i+1})) -(x(t_i)-\tilde{x}(t_i))  \|^p \right)^{1/p}.
We will then prove the following result:

Proposition: Let p < 2. If x^n:[0,1] \to \mathbb{R}^d is a sequence of bounded variation paths that converges in p-variation to a path x \in \Omega^p  (\mathbb{R}^d), then the solutions of the differential equations
y^n(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y^n(s)) dx^{i,n}(s),
converge in p-variation to some y \in \Omega^p  (\mathbb{R}^d). Moreover y is the solution of the differential equation
y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),
where the integrals are understood in the sense of Young’s integration.

The value p=2 is really a treshold: The result is simply false for p=2. The main idea of the rough paths theory is to introduce a much stronger topology than the convergence in p-variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups.

Let \mathbb{G}_N (\mathbb{R}^d) be the free N-step nilpotent Lie group with d generators X_1,\cdots,X_d. If x:[0,1] \rightarrow  \mathbb{R}^d is continuous with bounded variation, the solution x^* of the equation
x^*(t)=\sum_{i=1}^d \int_0^t X_i (x^*(s)) dx^i(s),
is called the lift of x in \mathbb{G}_N (\mathbb{R}^d). For p \geq 1, let us
denote \Omega^p \mathbb{G}_N (\mathbb{R}^d) the closure of the set of lifted paths x^*:[0,1]  \rightarrow \mathbb{G}_N (\mathbb{R}^d) with respect to the distance in p-variation which is given by
\delta^N_p (x^*,y^*) =\sup_{\pi} \left( \sum_{i=1}^{n-1} d_N \left( y_{t_i}^* (x_{t_i}^*)^{-1}, y_{t_{i+1}}^* (x_{t_{i+1}}^*)^{-1} \right)^p \right)^{\frac{1}{p}},
where d_N denotes the Carnot-Caratheodory distance on the group \mathbb{G}_N (\mathbb{R}^d). This is a distance that will be explained in details later. Its main property is that it is homogeneous with respect to the natural dilation of \mathbb{G}_N (\mathbb{R}^d).

Consider now the map \mathcal{I} which associates to a continuous with bounded variation path
x: [0,1] \rightarrow \mathbb{R}^d the continuous path with bounded variation y : [0,1] \rightarrow \mathbb{R}^d that solves the ordinary differential equation
y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).
It is clear that there exists a unique map \mathcal{I}^{*} from the set of continuous with bounded variation lifted paths [0,1] \rightarrow \mathbb{G}_N (\mathbb{R}^d) onto the set of continuous with bounded variation lifted paths [0,1] \rightarrow \mathbb{G}_N (\mathbb{R}^n) which makes the following diagram commutative
\begin{array}{lll}  & \mathcal{I}^{*} &  \\  x^{*} & \longrightarrow  & y^{*} \\  \uparrow  &  & \uparrow  \\  x & \longrightarrow  & y \\  & \mathcal{I} &  \end{array}.
The fundamental theorem of Lyons is the following:

Theorem: If N \geq [p], then in the topology of \delta^N_p-variation, there exists a continuous extension of \mathcal{I}^{*} from \Omega^p \mathbb{G}_N (\mathbb{R}^d) into \Omega^p \mathbb{G}_N (\mathbb{R}^n).

In particular, we can now give a sense to differential equations driven by some continuous paths with finite p-variation, for any p \ge 1. Indeed, let x:[0,1] \to \mathbb{R}^d which is continuous with a fnite p-variation and assume that there exists x^* \in \Omega^p \mathbb{G}_N (\mathbb{R}^d) whose projection onto \mathbb{R}^d is x. The projection onto \mathbb{R}^d of \mathcal{I}^{*}(x^*) is then understood as being a solution of
y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).

An important example of application is given by the case where the driving signal is a Brownian motion (B(t))_{t \ge 0}. Brownian motion has a p-finite variation for any p > 2 and, as we will see, admits a canonical lift in \Omega^p \mathbb{G}_2 (\mathbb{R}^d). As a conclusion, we can consider in the rough paths sense, solutions to the equation
y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dB^i(s).

It turns out that this notion of solution is exactly equivalent to solutions that are obtained by using the Stratonovitch integration theory. Therefore, the theory of stochastic differential equations appears as a very special case of the rough paths theory !

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