Show that if $W(t),\ 0 \le t \le 1$\ is a standard Wiener process, then its Holder norm $\sup\limits_{0 \le s,\ t \le 1}\frac{|W(t) - W(s)|}{|t - s|^{\alpha}},\ 0 < \alpha < \frac{1}{2}$\ has moments of all orders. Does the moment generating function exist?
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