## Lecture 8. Young’s differential equations

In the previous lecture we defined the Young’s integral $\int y dx$ when $x \in C^{p-var} ([0,T], \mathbb{R}^d)$ and $y \in C^{q-var} ([0,T], \mathbb{R}^{e \times d})$ with $\frac{1}{p}+\frac{1}{q} > 1$. The integral path $\int_0^t ydx$ has then a bounded $p$-variation. Now, if $V: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ is a Lipschitz map, then the integral, $\int V(x) dx$ is only defined when $\frac{1}{p}+\frac{1}{p} > 1$, that is for $p < 2$. With this in mind, it is apparent that Young’s integration should be useful to solve differential equations driven by continuous paths with bounded $p$-variation for $p < 2$. If $p \ge 2$, then the Young’s integral is of no help and the rough paths theory later explained is the correct one.

The basic existence and uniqueness result is the following. Throughout this lecture, we assume that $p < 2$.

Theorem: Let $x\in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e \times d}$ be a Lipschitz continuous map, that is there exists a constant $K > 0$ such that for every $x,y \in \mathbb{R}^e$,
$\| V(x)-V(y) \| \le K \| x-y \|.$
For every $y_0 \in \mathbb{R}^e$, there is a unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$
Moreover $y \in C^{p-var} ([0,T], \mathbb{R}^e)$.

Proof: The proof is of course based again of the fixed point theorem. Let $0 < \tau \le T$ and consider the map $\Phi$ going from the space $C^{p-var} ([0,\tau], \mathbb{R}^e)$ into itself, which is defined by
$\Phi(y)_t =y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le \tau.$
By using basic estimates on the Young’s integrals, we deduce that
$\| \Phi(y^1)-\Phi(y^2) \|_{ p-var, [0,\tau]}$
$\le C \| x \|_{p-var,[0,\tau]} ( \| V(y^1)-V(y^2) \|_{ p-var, [0,\tau]} +\| V(y^1)(0)-V(y^2)(0)\|)$
$\le CK \| x \|_{p-var,[0,\tau]}( \| y^1-y^2 \|_{ p-var, [0,\tau]}+\| y^1(0)-y^2(0)\|).$
If $\tau$ is small enough, then $CK \| x \|_{p-var,[0,\tau]} < 1$, which means that $\Phi$ is a contraction of the Banach space $C^{p-var} ([0,\tau], \mathbb{R}^e)$ endowed with the norm $\| y \|_{p-var,[0,\tau]} +\| y(0)\|$.

The fixed point of $\Phi$, let us say $y$, is the unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le \tau.$
By considering then a subdivision
$\{ \tau=\tau_1 < \tau_2 <\cdots <\tau_n=T \}$
such that $C K \| x \|_{p-var,[\tau_k,\tau_{k+1}]} < 1$, we obtain a unique solution to the differential equation:
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T$ $\square$

As for the bounded variation case, the solution of a Young's differential equation is a $C^1$ function of the initial condition,

Proposition: Let $x\in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e \times d}$ be a $C^1$ Lipschitz continuous map. Let $\pi(t,y_0)$ be the flow of the equation
$y(t)=y_0+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$
Then for every $0\le t \le T$, the map $y_0 \to \pi (t,y_0)$ is $C^1$ and the Jacobian $J_t=\frac{\partial \pi(t,y_0)}{\partial y_0}$ is the unique solution of the matrix linear equation
$J_t=Id+ \sum_{i=1}^d \int_0^t DV_i(\pi(s,y_0))J_s dx^i(s).$

As we already mentioned it before, solutions of Young’s differential equations are continuous with respect to the driving path in the $p$-variation topology

Theorem: Let $x^n \in C^{p-var} ([0,T], \mathbb{R}^d)$ and let $V : \mathbb{R}^e \to \mathbb{R}^{e\times d}$ be a Lipschitz and bounded continuous map such that for every $x,y \in \mathbb{R}^d$,
$\| V(x)-V(y) \| \le K \| x-y \|.$
Let $y^n$ be the solution of the differential equation:
$y^n(t)=y(0)+\int_0^t V(y^n(s)) dx^n(s), \quad 0\le t \le T.$
If $x^n$ converges to $x$ in $p$-variation, then $y^n$ converges in $p$-variation to the solution of the differential equation:
$y(t)=y(0)+\int_0^t V(y(s)) dx(s), \quad 0\le t \le T.$

Proof: Let $0\le s \le t \le T$. We have
$\| y-y^n \|_{p-var,[s,t]}$
$= \left\| \int_0^\cdot V(y(u)) dx(u) -\int_0^\cdot V(y^n(u)) dx^n(u) \right\|_{p-var,[s,t]}$
$\le \left\| \int_0^\cdot (V(y(u))-V(y^n(u))) dx(u) + \int_0^\cdot V(y^n(u)) d( x(u)-x^n(u)) \right\|_{p-var,[s,t]}$
$\le \left\| \int_0^\cdot (V(y(u))-V(y^n(u))) dx(u) \right\|_{p-var,[s,t]}+\left\| \int_0^\cdot V(y^n(u)) d( x(u)-x^n(u)) \right\|_{p-var,[s,t]}$
$\le CK \| x\|_{p-var,[s,t]} \| y-y^n \|_{p-var,[s,t]}+C\| x-x^n \|_{p-var,[s,t]}(K \| y^n \|_{p-var,[s,t]}+\| V\|_{\infty, [0,T]})$
Thus, if $s,t$ is such that $CK \| x\|_{p-var,[s,t]} < 1$, we obtain
$\| y-y^n \|_{p-var,[s,t]} \le \frac{C(K \| y^n \|_{p-var,[s,t]}+\| V\|_{\infty, [0,T]})}{ 1-CK\| x\|_{p-var,[s,t]} } \| x-x^n \|_{p-var,[s,t]}.$
In the very same way, provided $CK \| x^n\|_{p-var,[s,t]} < 1$, we get
$\| y^n \|_{p-var,[s,t]} \le \frac{C\| V\|_{\infty, [0,T]}}{ 1-CK\| x^n\|_{p-var,[s,t]} }.$

Let us fix $0 < \varepsilon < 1$ and pick a sequence $0\le \tau_1 \le \cdots \le \tau_m=T$ such that $CK \| x\|_{p-var,[\tau_i,\tau_{i+1}]}+\varepsilon < 1$. Since $\| x^n\|_{p-var,[\tau_i,\tau_{i+1}]} \to \| x\|_{p-var,[\tau_i,\tau_{i+1}]}$, for $n \ge N_1$ with $N_1$ big enough, we have
$CK \| x^n\|_{p-var,[\tau_i,\tau_{i+1}]}+\frac{\varepsilon}{2} < 1.$
We deduce that for $n \ge N_1$,
$\| y^n \|_{p-var,[\tau_i,\tau_{i+1}]} \le \frac{2}{\varepsilon} C \| V\|_{\infty, [0,T]}$
and
$\| y-y^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C(K \frac{2}{\varepsilon} C \| V\|_{\infty, [0,T]}+\| V\|_{\infty, [0,T]})}{ 1-CK\| x\|_{p-var,[\tau_i,\tau_{i+1}] }} \| x-x^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \| x-x^n \|_{p-var,[\tau_i,\tau_{i+1}]}$
$\le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \| x-x^n \|_{p-var,[0,T]}.$
For $n \ge N_2$ with $N_2 \ge N_1$ and big enough, we have
$\| x-x^n \|_{p-var,[0,T]} \le \frac{\varepsilon^3}{m},$
which implies
$\| y-y^n \|_{p-var,[0,T]} \le \frac{C}{\varepsilon} \| V\|_{\infty, [0,T]} \left( \frac{2KC}{\varepsilon}+1 \right) \varepsilon^3.$
$\square$

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