On some maximal inequalities for fractional Brownian motions
Let \(Z= (Z_t)_{t\geq 0}\) be a fractional Brownian motion with Hurst index \(H\). Denote by \(F= (F_t)_{t\geq 0}\) the filtration generated by \(Z\). Let \(\tau\) be a stopping time with respect to the filtration \(F\) and \(Z_t^*= \sup_{s\leq t} |Z_s|\). Then for any \(p>0\) and \(H\in (\frac 12,1)\) \[ c_1 E(\tau^{pH})\leq E((Z_\tau^*)^p)\leq c_2 E(\tau^{pH}), \] and for any \(p>0\) and \(H\in (0, \frac 12)\) \[ c_1 E(\tau^{pH})\leq E((Z_\tau^*)^p), \] where the constants \(c_1,c_2> 0\) depend only on the parameters \(p\), \(H\).
scientific article; zbMATH DE number 1395118