An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion

Coutin, Laure

doi:10.1007/978-3-540-71189-6_1

Cited by 46 publications

(49 citation statements)

References 67 publications

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“…As a consequence, the construction for the fBm of a stochastic calculus turns out to be more involved than for the cBm. This can be done by several way [7], and here we use the rough paths approach as in [8].…”

Section: Rough Pathsmentioning

confidence: 99%

Acoustic Waves in Long Range Random Media

Marty¹,

Sølna²

2009

SIAM J. Appl. Math.

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Abstract. We consider waves propagating through multiscale media. Much is known about waves propagating through a medium that satisfies a scale separation assumption with random fluctuations on a microscale. Here we go beyond this situation and consider waves propagating through a medium defined in terms of a long range process. Such a medium can, for instance, be modeled in terms of a one-dimensional fractional Brownian motion with variations on a continuum of scales. Fractal medium models are used to model, for example, the heterogeneous earth and the turbulent atmosphere. We set forth a framework using the theory of rough paths in which propagation problems of this nature can be analyzed in the case with anticipative medium fluctuations with a Hurst exponent H > 1/2. We show how the wave interacts with the medium fluctuations in this case and that the interaction is qualitatively different from the situation where the medium satisfies a separation of scales assumption. In the long range case considered here the travel time depends strongly on the particular medium realization, but in fact the pulse shape does not.Key words. wave propagation, random media, long range processes, fractional Brownian motion AMS subject classifications. 34F05, 34E10, 37H10, 60H20 DOI. 10.1137/07068610X1. Introduction. Modeling in terms of a multiscale medium is important for propagation problems in, for instance, the earth's crust, the turbulent atmosphere, turbulent boundary layers, sea ice, and outer space [9,15,19,20,34,42,44]. Communication, remote sensing, and laser beam propagation schemes are affected by bad weather and multiscale medium variations. Large scale research projects (the ABLE ACE program Kirtland AFB, for instance [43]) have focused on gathering atmospheric turbulence data and numerically simulating propagation of wave fields through synthetic turbulence models that derive from these. Above a boundary layer atmospheric turbulence may occur within a stratified environment, and the turbulent temperature variations may be highly anisotropic; see [16,36]. Rough and long range medium fluctuations associated with multiscale modeling are also important in medical imaging, device modeling, and nuclear technology, to name a few. The zone in between different tissue types (or in between different dielectrica) may in particular be strongly heterogeneous with variation on a continuum of scales. In general, the detailed pointwise variation of a multiscale medium, the refractive index, say, cannot be identified. However, the statistics of this variation can be characterized. Optimal design of, for instance, imaging and communication algorithms requires insight about how the wave is affected by the rough medium fluctuations, that is, the nonlinear coupling between medium and wave field statistics. This is particularly the case with modern high resolution sampling and imaging technology. Insight about the wave medium interaction is also important in a range of other applications like design of sound-absorbing materials and non...

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Section: Rough Pathsmentioning

confidence: 99%

Acoustic Waves in Long Range Random Media

Marty¹,

Sølna²

2009

SIAM J. Appl. Math.

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“…We may then recover some of these results: existence of solutions of such equations has been provided in [31]. The properties of such equations are studied for example in [1,5,6,8,14,15,[25][26][27][28][29][30]32,35] . .…”

Section: Introductionmentioning

confidence: 95%

“…For this, several approaches have been proposed (see [5,24] for a survey), and one of them relies on the rough paths theory.…”

Section: Introductionmentioning

confidence: 99%

Controlled differential equations as Young integrals: A simple approach

Lejay

2010

Journal of Differential Equations

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The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1 p < 2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory -existence, uniqueness, convergence of the Euler scheme, flow property . . . -which are spread out among several articles.

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“…In the last few decades, many differential ways have been introduced to constructed the fractional stochastic calculus (see, for instance, [30]). The main difficulties in studying fractional stochastic systems are that we cannot apply stochastic calculus developed by Itô since fBm is neither a Markov process nor a semimartingale, except for H = 1/2.…”

Section: Preliminariesmentioning

confidence: 99%

Harnack Inequality and Applications for Stochastic Retarded Differential Equations Driven by Fractional Brownian Motion

Liu¹

2017

JPDE

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Abstract. In this paper, by using a semimartingale approximation of a fractional stochastic integration, the global Harnack inequalities for stochastic retarded differential equations driven by fractional Brownian motion with Hurst parameter 0 < H < 1 are established. As applications, strong Feller property, log-Harnack inequality and entropycost inequality are given. AMS Subject Classifications: 60G15, 60H05, 60H15Chinese Library Classifications: O211.63

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An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion

Cited by 46 publications

References 67 publications

Acoustic Waves in Long Range Random Media

Acoustic Waves in Long Range Random Media

Controlled differential equations as Young integrals: A simple approach

Harnack Inequality and Applications for Stochastic Retarded Differential Equations Driven by Fractional Brownian Motion

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