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Fractional Brownian motion [PDF]

, 2006
Fractional Brownian motion is a stochastic process which deviates significantly from Brownian motion and semimartingales, and others classically used in probability theory. As a centered Gaussian process, it is characterized by the stationarity of its increments and a medium-or long-memory property which is in sharp contrast with martingales and Markov
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The renormalization group and fractional Brownian motion [PDF]

Physics Letters A, 2002
We find that in generic field theories the combined effect of fluctuations and interactions leads to a probability distribution function which describes fractional Brownian Motion (fBM) and ``complex behavior''. To show this we use the Renormalization Group as a tool to improve perturbative calculations, and check that beyond the classical regime of ...
Juan Pérez-Mercader, David Hochberg
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Are Fractional Brownian Motions Predictable? [PDF]

, 2011
We provide a device, called the local predictor, which extends the idea of the predictable compensator. It is shown that a fBm with the Hurst index greater than 1/2 coincides with its local predictor while fBm with the Hurst index smaller than 1/2 does not admit any local predictor.
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Ball throwing on spheres [PDF]

, 2009
Ball throwing on Euclidean spaces has been considered for a while. A suitable renormalization leads to a fractional Brownian motion as limit object. In this paper we investigate ball throwing on spheres.
Estrade, Anne, Istas, Jacques
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Fractional Brownian motion in a nutshell [PDF]

International Journal of Modern Physics: Conference Series, 2015
This is an extended version of the lecture notes to a mini-course devoted to fractional Brownian motion and delivered to the participants of the 7th Jagna International Workshop.
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Asymptotic Growth of Sample Paths of Tempered Fractional Brownian Motions, with Statistical Applications to Vasicek-Type Models

Fractal and Fractional
Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering.
Yuliya Mishura, Kostiantyn Ralchenko
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Arbitrage with Fractional Brownian Motion [PDF]

Mathematical Finance, 1997
Fractional Brownian motion has been suggested as a model for the movement of log share prices which would allow long–range dependence between returns on different days. While this is true, it also allows arbitrage opportunities, which we demonstrate both indirectly and by constructing such an arbitrage.
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A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter [PDF]

, 2014
Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional ...
Richard, Alexandre
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Extremes of spherical fractional Brownian motion [PDF]

Extremes, 2019
Let $\{B_ (x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $ $. We study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_ (x) > u \}$ and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere $\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$
Cheng, Dan, Liu, Peng
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Option Pricing under the Subordinated Market Models

Discrete Dynamics in Nature and Society, 2022
This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity ...
Longjin Lv, Changjuan Zheng, Luna Wang
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