Fractional Brownian motion satisfies two-way crossing

Rémi Peyre (1)

1 IECL - Institut Élie Cartan de Lorraine

We prove the following result: For (Z_t)_{t∈R} a fractional Brownian motion with arbitrary Hurst parameter, for any stopping time τ, there exist arbitrarily small ε > 0 such that Z_{τ+ε} < Z_τ, with asymptotic behaviour when ε ↘ 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.

Mots clés: fractional Brownian motion; stopping time; law of the iterated logarithm; two-way crossing

Domaines: Probabilités [math.PR]

Rémi Peyre. Fractional Brownian motion satisfies two-way crossing. Bernoulli, 2017, ⟨10.3150/16-BEJ858⟩. ⟨hal-01282229⟩