satisfies $\left|  \frac{\partial K(t, s)}{\partial t} \right| \leq C \left(  t-s \right)^{-(\alpha + 1)}$.

The following proposition summarizes the properties of fractional Brownian motion which will be relevant in the sequel.

\[fBMCond\] Let $B^{H}$ be standard fractional Brownian motion with Hurst index $H\in\left(  \frac{1}{4},1\right)  $, and let $K$ be the square-integrable kernel associated with it. We have:

(i) For any $p>\frac{1}{H}$ the sample paths of $B^{H}$ are almost surely $\frac{1}{p}$-Hölder continuous. Furthermore, there exists a geometric rough path $\mathbf{X}\in\mathcal{C}^{0,p-var}\left( [0,T];G^{\lfloor p\rfloor}\left(  \mathbb{R}^{d}\right)  \right) $ which is the $d_{p\text{-var }}$-limit of the paths $S_{\left\lfloor p\right\rfloor }\left(  X^{\pi}\right)  $ as $\left\vert \left\vert \pi\right\vert \right\vert \rightarrow 0$.

(ii) $B^{H}$ satisfies Condition \[newCond1\] with $\rho=\frac{1}{2H}$ and $$\begin{aligned}
    q = \begin{cases}
    \: \frac{2\rho}{\rho+1} \quad \text{if }\frac{1}{4}<H\leq\frac{1}{3},\\
    \: \frac{1}{\rho}\wedge1 \quad \text{if }\frac{1}{3} < H \leq 1.
    \end{cases}\end{aligned}$$

(iii) If $\frac{1}{4} < H \leq \frac{1}{2}$, then the kernel $K$ satisfies Condition \[amnCond\] with $\alpha = \frac{1}{2} - H$.

(iv) The covariance function (\[covfbm\]) of $B^{H}$ satisfies:

    1.  $\left\Vert R(t,\cdot)-R(s,\cdot)\right\Vert _{q-var;[0,T]}
        \leq C \left\vert t-s\right\vert ^{\frac{1}{\rho}}$, if $\frac{1}{4}<H\leq\frac{1}{2}$,

    2.  $R(t)=t^{2H}$ is of bounded variation and thus of finite q-variation for any $q \geq 1$.

The sample paths of fractional Brownian motion have $H - \varepsilon$-Hölder regularity for any 