The Kolmogorov Continuity theorem (see for example the [Wikipedia page](https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem)) lets us prove that a stochastic process $X\_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided we have a bound of the form
$$\mathbb E\left[d(X\_t,X\_s)^\alpha\right]\le K |t-s|^{1+\beta}$$
where $\alpha, \beta>0$. More precisely, it shows that there exists a random variable $c>0$ such that
$$d(X\_t,X\_s)\le c|t-s|^\gamma $$
where $\gamma\in (0,\frac\beta\alpha)$.


However, this doesn't seem to give any information on the Holder constant $c$. I am wondering if there is some relation between $K$ and $c$. For example, is it possible to estimate the moments
$$\mathbb E[c^n]$$
in terms of $K$?


This seems to be related to this [unanswered question](https://mathoverflow.net/questions/95656/moment-bounds-on-h%C3%B6lder-norms-of-stochastic-processes) many years ago.



Kolmogorov continuity theorem and Holder norm
One can apply a deterministic result, called Garsia--Rodemich--Rumsey inequality, to estimate $\mathrm{E}[||X||^\alpha\_{\gamma;[0,T]}]$. Here is a particular form of this result, which is most convenient for us.



> 
> For any $\alpha >1$, $\delta> 1/\alpha$, there is a constant $C(\delta,\alpha)$ such that for any $f\in C([0,T],S)$ and $t,s\in[0,T]$
>  $$
> d(f(t), f(s))^\alpha\le C(\delta,\alpha)|t-s|^{\delta\alpha - 1} \int\_s^t \int\_s^t \frac{d(f(u),f(v))^\alpha}{|u-v|^{\delta\alpha + 1}}du\,dv.
> $$
>  In particular, for $\gamma = \delta - 1/\alpha$
>  $$
> ||f||\_{\gamma;[0,T]}:= \sup\_{0\le t<s\le T}\frac{d(f(t),f(s))}{|t-s|^\gamma}\le C(\delta,\alpha)^{1/\alpha}\bigg(\int\_0^T\int\_0^T \frac{d(f(u),f(v))^\alpha}{|u-v|^{\delta\alpha + 1}}du\,dv\bigg)^{1/\alpha}.
> $$
> 
> 
> 


For any $\gamma<\beta/\alpha$ take $\delta\in (1/\alpha,\gamma+1/\alpha)$ to get 
$$
\mathrm E\left[||X||\_{\gamma;[0,T]}^\alpha\right]\le C(\delta,\alpha) \int\_0^T\int\_0^T \frac{\mathrm{E} \left[d(X\_u,X\_v)^\alpha\right]}{|u-v|^{\delta\alpha + 1}}du\,dv \\
 \le C(\delta,\alpha) K\int\_0^T\int\_0^T |u-v|^{\beta-\delta\alpha}du\,dv = C(\delta,\alpha,\beta,T)K.
$$
As a result,
$$
\mathrm E\left[||X||\_{\gamma;[0,T]}^\alpha\right]\le C(\alpha,\beta,\gamma,T) K,
$$
and the constant may be written explicitly.


This is slightly better than @Kostya\_I's estimate $\mathrm{P}(||X||\_{\gamma;[0,T]}^\alpha > x)\le CKx^{-1}$, which does not imply integrability. 


