In this lecture we introduce the central notion of the signature of a path which is a convenient way to encode all the algebraic information on the path which is relevant to study differential equations driven by . The motivation for the definition of the signature comes from formal manipulations on Taylor series.

Let us consider a differential equation

where the ‘s are smooth vector fields on .

If is a function, by the change of variable formula,

Now, a new application of the change of variable formula to leads to

We can continue this procedure to get after steps

for some remainder term , where we used the notations:

- If is a word with length ,

If we let , assuming (which is by the way true for small enough if the ‘s are analytic), we are led to the formal expansion formula:

This shows, at least at the formal level, that all the information given by on is contained in the iterated integrals .

Let be the non commutative algebra over of the formal series with indeterminates, that is the set of series

**Definition:** *Let . The signature of (or Chen’s series) is the formal series:
*

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 . For any word and any ,
where we used the convention that if is a word with length 0, then . *

**Proof:** It follows readily by induction on by noticing that

To avoid heavy notations, it will be convenient to denote

This notation actually reflects a natural algebra isomorphism between and . With this notation, observe that the signature writes then

and that the Chen’s relations become

The Chen’s relations imply the following flow property for the signature:

**Proposition:*** Let . For any ,
*

**Proof:** Indeed,