## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3571-3597.

### Fractional Brownian motion satisfies two-way crossing

Rémi Peyre

#### Abstract

We prove the following result: For $(Z_{t})_{t\in\mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, for any stopping time $\tau$, there exist arbitrarily small $\varepsilon>0$ such that $Z_{\tau+\varepsilon}<Z_{\tau}$, with asymptotic behaviour when $\varepsilon\searrow0$ 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.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3571-3597.

Dates
Revised: February 2016
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bj/1495505102

Digital Object Identifier
doi:10.3150/16-BEJ858

Mathematical Reviews number (MathSciNet)
MR3654816

Zentralblatt MATH identifier
06778296

#### Citation

Peyre, Rémi. Fractional Brownian motion satisfies two-way crossing. Bernoulli 23 (2017), no. 4B, 3571--3597. doi:10.3150/16-BEJ858. https://projecteuclid.org/euclid.bj/1495505102

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