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November 2017 Fractional Brownian motion satisfies two-way crossing
Rémi Peyre
Bernoulli 23(4B): 3571-3597 (November 2017). DOI: 10.3150/16-BEJ858

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.

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Rémi Peyre. "Fractional Brownian motion satisfies two-way crossing." Bernoulli 23 (4B) 3571 - 3597, November 2017. https://doi.org/10.3150/16-BEJ858

Information

Received: 1 September 2015; Revised: 1 February 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 06778296
MathSciNet: MR3654816
Digital Object Identifier: 10.3150/16-BEJ858

Keywords: fractional Brownian motion , Law of the iterated logarithm , stopping time , two-way crossing

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 4B • November 2017
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