Iterated Integrals and Exponential Homomorphisms^†

Differential Equations Driven by Rough Paths

“…For each p ∈ [1,2) and each integer n ≥ 0, the truncated signature is a continuous mapping The element π n (S(X)) is called the truncated signature of X of order n. Corollary 2.11.…”

“…According to the rules defining the tensor product in T (2) (V ) (see Definition 2.5), the identities satisfied by X s,t are the following: for all 0 ≤ s ≤ u ≤ t ≤ T , According to the rules defining the tensor product in T (2) (V ) (see Definition 2.5), the identities satisfied by X s,t are the following: for all 0 ≤ s ≤ u ≤ t ≤ T ,…”

“…For instance, if w is any non-zero tensor in V ⊗2 , then the functional X s,t = (1, 0, (t − s)w) ∈ T (2) (V ) is a multiplicative functional of degree 2. Hence, by the remark made earlier, there exists a classical path x : [0, T ] −→ V which underlies X in the sense that X 1 s,t = x t − x s for all (s, t) ∈ T .…”

“…In this particular case, we can identify G (2) in very concrete terms. In fact for any two indices i and j, the pair x i u , x j u defines a directed planar curve.…”

“…This was discovered by Young around 1930 and allows one to make sense of and even, if f is regular enough, to solve (2) when X has finite p-variation for p < 2. Moreover, in this case, the integral, as a function of t, has finite p-variation, like X.…”