## Lecture 9. The signature of a bounded variation path

In this lecture we introduce the central notion of the signature of a path $x \in C^{1-var}([0,T],\mathbb{R}^d)$ which is a convenient way to encode all the algebraic information on the path $x$ which is relevant to study differential equations driven by $x$. The motivation for the definition of the signature comes from formal manipulations on Taylor series.

Let us consider a differential equation
$y(t)=y(s)+\sum_{i=1}^d \int_s^t V_i (y(u) )dx^i(u),$
where the $V_i$‘s are smooth vector fields on $\mathbb{R}^n$.

If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a $C^{\infty}$ function, by the change of variable formula,
$f(y(t))=f(y(s))+\sum^{d}_{i=1}\int^{t}_{s}V_{i}f(y(u))dx^{i}(u).$

Now, a new application of the change of variable formula to $V_{i}f(y(s))$ leads to
$f(y(t))=f(y(s))+\sum^{d}_{i=1}V_{i}f(y(s))\int^{t}_{s}dx^{i}(u)+\sum^{d}_{i,j=1}\int^{t}_{s}\int^{u}_{s} V_{j}V_{i}f(y(v))dx^{j}(v)dx^{i}(u).$

We can continue this procedure to get after $N$ steps
$f(y(t))=f(y(s))+\sum^{N}_{k=1}\sum_{I=(i_1,\cdots,i_k)}(V_{i_1}\cdots V_{i_k}f)(y(s))\int_{\Delta^{k}[s,t]}dx^{I}+R_{N}(s,t)$
for some remainder term $R_{N}(s,t)$, where we used the notations:

• $\Delta^{k}[s,t]=\{(t_1,\cdots,t_k)\in[s,t]^{k}, s\leq t_1\leq t_2\cdots\leq t_k\leq t\}$
• If $I=\left(i_1,\cdots,i_k\right)\in\{1,\cdots,d\}^k$ is a word with length $k$, $\int_{\Delta^{k}[s,t]}dx^{I}=\displaystyle \int_{s \le t_1 \le t_2 \le \cdots \le t_k \le t}dx^{i_1}(t_1)\cdots dx^{i_k}(t_k).$

If we let $N\rightarrow +\infty$, assuming $R_{N}(s,t) \to 0$ (which is by the way true for $t-s$ small enough if the $V_i$‘s are analytic), we are led to the formal expansion formula:
$f(y(t))=f(y(s))+\sum^{+\infty}_{k=1}\sum_{I=(i_1,\cdots,i_k)}(V_{i_1}\cdots V_{i_k}f)(y(s))\int_{\Delta^{k}[s,t]}dx^{I}.$
This shows, at least at the formal level, that all the information given by $x$ on $y$ is contained in the iterated integrals $\int_{\Delta^{k}[s,t]}dx^{I}$.

Let $\mathbb{R} [[X_1,...,X_d]]$ be the non commutative algebra over $\mathbb{R}$ of the formal series with $d$ indeterminates, that is the set of series
$Y=y_0+\sum_{k = 1}^{+\infty} \sum_{I \in \{1,...,d\}^k} a_{i_1,...,i_k} X_{i_1}...X_{i_k}.$

Definition: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. The signature of $x$ (or Chen’s series) is the formal series:
$\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \sum_{I \in \{1,...,d\}^k} \left( \int_{\Delta^{k}[s,t]}dx^{I} \right) X_{i_1} \cdots X_{i_k}, \quad 0 \le s \le t \le T.$

As we are going to see in the next few lectures, the signature is a fascinating algebraic object. At the source of the numerous properties of the signature lie the following so-called Chen’s relations

Lemma: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. For any word $(i_1,...,i_n) \in \{ 1, ... , d \}^n$ and any $0 \le s \le t \le u \le T$,
$\int_{\Delta^n [s,u]} dx^{(i_1,...,i_n)}=\sum_{k=0}^{n} \int_{\Delta^k [s,t]} dx^{(i_1,...,i_k)}\int_{\Delta^{n-k} [t,u]} dx^{(i_{k+1},...,i_n)},$
where we used the convention that if $I$ is a word with length 0, then $\int_{\Delta^{0} [0,t]} dx^I =1$.

Proof: It follows readily by induction on $n$ by noticing that
$\int_{\Delta^n [s,u]} dx^{(i_1,...,i_n)}=\int_s^u \left( \int_{\Delta^{n-1} [s,t_n]} dx^{(i_1,...,i_{n-1})} \right) dx^{i_n}(t_n)$ $\square$

To avoid heavy notations, it will be convenient to denote
$\int_{\Delta^k [s,t]} dx^{\otimes k} =\sum_{I \in \{1,...,d\}^k} \left( \int_{\Delta^{k}[s,t]}dx^{I} \right) X_{i_1} \cdots X_{i_k}.$

This notation actually reflects a natural algebra isomorphism between $\mathbb{R} [[X_1,...,X_d]]$ and $1\oplus_{k=1}^{+\infty} (\mathbb{R}^d)^{\otimes k}$. With this notation, observe that the signature writes then
$\mathfrak{S} (x)_{s,t} =1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,t]} dx^{\otimes k},$
and that the Chen’s relations become
$\int_{\Delta^n [s,u]} dx^{\otimes n}=\sum_{k=0}^{n} \int_{\Delta^k [s,t]} dx^{\otimes k }\int_{\Delta^{n-k} [t,u]} dx^{\otimes (n-k) }.$
The Chen’s relations imply the following flow property for the signature:

Proposition: Let $x \in C^{1-var}([0,T],\mathbb{R}^d)$. For any $0 \le s \le t \le u \le T$,
$\mathfrak{S} (x)_{s,u} =\mathfrak{S} (x)_{s,t}\mathfrak{S} (x)_{t,u}$

Proof: Indeed,
$\mathfrak{S} (x)_{s,u}$
$=1 + \sum_{k=1}^{+\infty} \int_{\Delta^k [s,u]} dx^{\otimes k}$
$=1 + \sum_{k=1}^{+\infty}\sum_{j=0}^{n} \int_{\Delta^j [s,t]} dx^{\otimes j }\int_{\Delta^{k-j} [t,u]} dx^{\otimes (k-j) }$
$=\mathfrak{S} (x)_{s,t}\mathfrak{S} (x)_{t,u}$
$\square$

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