Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 31 / 1 / i / Solution

Use the truncated tensor algebra $T^{(N)}({\mathbb{R}}^d)={⨁}_{k=0}^N({\mathbb{R}}^d{)}^{\otimes k}$, with truncated concatenation product. The step-$N$ path signature isThese iterated integrals of a path exist as Riemann-Stieltjes integrals, since a Lipschitz continuous path on a compact interval has bounded variation. The degree-zero coordinate is $1$, and the degree-one coordinate is $x_t-x_s$.

Let $S_t=S_N(x{)}_{0,t}$ and write $S_t^{(k)}$ for its degree-$k$ coordinate. Integrating first over all variables except the last givesThus $d{S}_t^{(k)}=S_t^{(k-1)}\otimes d{x}_t$ for $1\le k\le N$. If $e_i$ is a standard basis vector, define the linear vector field $W_i(g)=g\otimes e_i$, with degrees above $N$ discarded. The component identities are precisely the controlled ordinary differential equationFor a Lipschitz driver this may also be written ${\dot{S}}_t=S_t\otimes {\dot{x}}_t$ almost everywhere. Recursive integration of its levels proves uniqueness and recovers the signature formula. The Chen identity follows by splitting each integration simplex at an intermediate time.