Fractional Brownian motion satisfies two-way crossing

We prove the following result: For (Z(t))t is an element of R a fractional Brownian motion with arbitrary Hurst parameter, for any stopping timer, there exist arbitrarily small epsilon > 0 such that Zeta(tau+epsilon) < Z(tau), with asymptotic behaviour when epsilon SE arrow 0 satisfying a bound of iterated logarithm type. As a consequence, fractional Brownian motion satisfies the "two-way crossing" property, which has important applications in financial mathematics.