Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Theorem 12.5 to show that \(\kappa(t)=(1+\epsilon) \sqrt{2 t \log |\log t|}\) is an upper function for \(t ightarrow 0\).

Data From Theorem 12.5

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12.5 Theorem (Kolmogorov's test). Let (Br)o be a BM and x: (0,1] (0,00) continuous function such that x(t) is increasing and x(t)/f is decreasing. Then the following alternative holds: Ex. 12.4 Ex. 12.5 x(t) exp(-) dt 1)=1. B(t) x(t) XI 1) = 1. (12.7)