Fractional Brownian motion: Small increments and first exit time from one-sided barrier

Let {BH(t)}t≥0 be a fractional Brownian motion indexed by non-negative time with initial value BH(0)=0 almost surely and Hurst parameter H∈(0,1). By using a random wavelet series representation of {BH(t)}t≥0, we show that the fractional Brownian increment |BH(t+h)−BH(t)| is almost surely bounded from above by C|h|Hloglog|h|−1 when the time variation |h| is sufficiently small, where the random variable C>0 does not depend on t nor on h; and an upper bound of the p’th moment of C, E[Cp], is provided for each p>0. This result fills some gap in the law of iterated logarithm for fractional Brownian motion, by giving the moments’ control of the almost sure upper bound of fractional Brownian increments. With this enhanced upper bound and some new results on the distribution of the maximum of fractional Brownian motion maxt∈[0,T]BH(t), we obtain a new and refined asymptotic estimate of the upper-tail probability P(τb>T) as T→+∞, where τb is the waiting time for {BH(t)}t≥0 (with H<1/2) to first exit from a positive-valued barrier b.