The Kolmogorov Continuity theorem (see for example the Wikipedia page) 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 many years ago.