Use Theorem 10.6 to show that the strong \(p\)-variation \(\operatorname{VAR}_{p}(B ;[0,1])\) of \(\mathrm{BM}^{1}\) is for \(p>2\) finite.

Data From Theorem 10.6

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10.6 Theorem (Lvy 1937). Let (B)o be a BM. Then P suposts-h B(t + h) - B(t)| 2h|log h (0-1)=1. Proof. We split the proof into two steps. We begin with the easier part and show "lim...>1". Since the limes superior is the largest accumulation point it is enough to consider a particular sequence h = 2 0. In fact, lim suposts 1- B(t + h) - B(t)| >1>lim 2h| log h maxoje2-1B()-B() 2-2-"log 2n maxo2-1B()-B(). = lim reQ(0,1), /2-2-log 2 -A,