A law of iterated logarithm for the subfractional Brownian motion and an application

Abstract

Abstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 00 $t > 0$, 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 $\Phi_{H}(0)=0$.

Keywords

• Sub-fractional Brownian motion
• Iterated logarithm
• Φ-variation