I know that the fractional Brownian motion has a Holder-continuous version (thanks to Kolmogorov's continuity theorem for example). Basically, for any $T>0$, $\epsilon>0$ and $t,s \in [0,T]$, there exists a random variable, say $C_T$ such that: $$ \mid B_t - B_s \mid \leq C_T \mid t - s \mid^{H-\epsilon} \text{ a.s}$$
Say $H$ lives in a compact set of $(0,1)$ and therefore we can make the constant $C_T$ independent of $H$.
I am interested in the dependence of $C_T$ with respect to $T$. I realize that it shouldn't grow too fast with $T$ but I couldn't find somewhere where it is explicitly given.
On the other hand, I am also interested in the regularity of $B$ with respect to $H$. That is with similar arguments (Kolmogorov), one can show that for $t \in [0,T]$ and $H_1,H_2$ in a compact set of $(0,1)$, we have: $$\mid B_t^{H_1}- B_t^{H_2} \mid \leq C_t \mid H_1 - H_2 \mid^\alpha \text{ a.s}$$ But again, how does $C_t$ depends on $T$?
