Is the Hölder random constant of the Brownian Motion Integrable?

Question

Let $\{B_t:t\in [0,1]\}$ be the standard one-dimensional Brownian motion on the closed unit interval. Fix $\gamma\in (0,1/2)$. It is well known that there is a positive random variable $K\equiv K(\gamma)$ such that for any pair $s,t\in [0,1]$ we have $$ |B_t-B_s|\leq K|t-s|^{\gamma} \qquad \text{a.s.} $$ I would like to know if $K$ can be chosen so that $\mathbb{E}[K]<+\infty$.

Answer 1

In the text Brownian Motion by R. Schilling L. Partzsch, one can see a demonstration of the fact that, for fixed $\gamma\in(0,1/2)$, the random variable $$ K:=\sup\{|B_t-B_s|/|t-s|^\gamma: 0\le s,t\le 1\} $$ has finite moments of all orders; see Theorem 10.1 on page 150.