# Levy modulus of continuity for a martingale

Given a Brownian motion $$B(t)$$ then (Levy, 1937) $$$$\mathbf{P}\bigg(\lim_{h\rightarrow 0}\frac{\sup_{0\le t\le 1-h}|B(t+h)-B(t)|}{ \sqrt{2hlog(1/h)}}=1\bigg)=1$$$$ Can the result (or a similar result) still hold for a martingale $$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}$$ or does one need a quadratic variation $$_{t}=t$$.