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,

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}

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