The proof of Theorem 10.3 actually shows, that almost all Brownian paths are nowhere Lipschitz continuous. Modify the argument of this proof to show that almost all Brownian paths are nowhere Hölder continuous for all \(\alpha>1 / 2\). Why does the argument break down for \(\alpha=1 / 2\) ?

It is enough to consider rational \(\alpha\) since \(\alpha\)-Hölder continuity implies \(\beta\)-Hölder continuity for all \(\beta

Data From Theorem 10.3

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10.3 Theorem (Paley, Wiener, Zygmund 1931). Let (Bt)to be a BM. Then the sample path t B+(w) is for almost all we nowhere differentiable. Proof (Dvoretzky, Erds, Kakutani 1961).. Set for every n > 1 == An = {we B(, w) nowhere differentiable in [0, n)}. It is not clear if the set A, is measurable. We will show that Q\An c Nn for a measurable null set Nn. Assume that the function w is differentlable at to = [0, n). Then te B(to, 6) w(t) - w(to)| < Llt-tol. : Consider for sufficiently large values of k > 1 the grid {j= 1,...,nk}. Then there exists a smallest index j = j(k) such that 1 to < and ...., +3 B (to, 6). Fori j+1,1+2,1+3 we get therefore k w()-w()w()-w(to)| + |w(to) - w()| L(-to-tol)