Rough path

Remarks

The map $\mathbb{X}_{s,t}$ stands for the iterated integral $\int_s^t X_{s,r} \otimes dX_r$. Clause 2 says that $X$ is Hölder continuous with exponent $\alpha$. Example: for a continuously differentiable path $X$, set $\mathbb{X}_{s,t} = \int_s^t X_{s,r} \otimes \dot{X}_r \, dr$. Splitting $\int_s^t$ at $u$ and writing $X_{s,r} = X_{s,u} + X_{u,r}$ for $r \ge u$ gives Chen's relation, and clauses 2 and 3 hold with exponents $1$ and $2$. Example: for a Brownian motion $B$, the Itô integrals $\mathbb{B}_{s,t} = \int_s^t B_{s,r} \otimes dB_r$ give a rough path almost surely, for each $\alpha \in (1/3, 1/2)$. Non-example: a smooth path $X$ with $\mathbb{X} = 0$ fails Chen's relation when $X_{s,u} \otimes X_{u,t} \ne 0$. The second level is not determined by $X$: if $g \colon [0, T] \to \R^d \otimes \R^d$ is $2\alpha$-Hölder, then $\mathbb{X}_{s,t} + g_t - g_s$ also satisfies the three clauses.