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Given a Brownian motion $B(t)$ then (Levy, 1937) \begin{equation} \mathbf{P}\bigg(\lim_{h\rightarrow 0}\frac{\sup_{0\le t\le 1-h}|B(t+h)-B(t)|}{ \sqrt{2hlog(1/h)}}=1\bigg)=1 \end{equation} Can the result (or a similar result) still hold for a martingale $M(t)$ replacing $B(t)$ such that $M(t)=\int_{0}^{t}f(s)dB(s)$ or $M(t)=\int_{0}^{t}f(M(s))dB(s)$ with some function $f:\mathbb{R}\rightarrow\mathbb{R}$ or does one need a quadratic variation $<M>_{t}=t$.

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