| Summary: | Abstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 0<H<1 $0< H<1$. In this paper, we give a local law of the iterated logarithm of the form lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, $$\limsup_{s\downarrow0}\frac{ \vert S^{H}_{t+s}-S^{H}_{t} \vert }{ s^{H}\sqrt {2\log^{+}\log(1/s)}}=1, $$ almost surely, for all t>0 $t > 0$, where log+x=max{1,logx} $\log^{+}x=\max{\{1, \log x\}}$ for x≥0 $x\geq0$. As an application, we introduce the ΦH $\Phi_{H}$-variation of SH $S^{H}$ driven by ΦH(x):=[x/2log+log+(1/x)]1/H $\Phi_{H}(x):= [x/\sqrt{2\log^{+}\log ^{+}(1/x)} ]^{1/H}$ (x>0) $(x>0)$ with ΦH(0)=0 $\Phi_{H}(0)=0$.
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