Signature of a bounded variation path ($S_N(x{)}_{s,t}=\left(1,{\int }_s^{t}dx,\dots ,{\int }_{s<u_1<\cdots <u_N<t}d{x}_{u_1}\otimes \cdots \otimes d{x}_{u_N}\right)$)

The path signature collects the ordered iterated Riemann-Stieltjes integrals of a continuous path of bounded variation. Its truncation belongs to the free step-N nilpotent Lie group. It multiplies under concatenation by the Chen identity, is independent of orientation-preserving reparametrisation of the whole path, and records more than its endpoint displacement.

For a continuous path of bounded variation, integrate successively over the ordered simplex using Riemann-Stieltjes integrals. These integrals form the levels of its path signature. Appending the final integration variable gives the differential recursion $d{S}_t^{(k)}=S_t^{(k-1)}\otimes d{x}_t$.

The product of two coordinate iterated integrals of a path is the sum over order-preserving interleavings of their integration variables. Split the product of the two ordered simplices according to the joint order of all variables. This identity explains the algebraic constraints on a geometric rough path and the cancellation of low-degree terms in the composition of its local Euler maps.

Partition each ordered integration simplex according to the number of times before the concatenation point. The integral over each resulting region factors into an initial iterated integral and a final iterated integral. Summing these factorizations proves the identity in the truncated tensor algebra.