On the representation of fractional brownian motion as an integral with respect to (dt) a
TLDR
In which way this notation can be extended to Brownian motion of fractional order a (different from 1/2) defined as the Riemann–Liouville derivative of the Gaussian white noise is examined.About:
This article is published in Applied Mathematics Letters.The article was published on 2005-07-01 and is currently open access. It has received 186 citations till now. The article focuses on the topics: Fractional Brownian motion & Stochastic differential equation.read more
Citations
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Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results
TL;DR: A modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's.
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Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions
TL;DR: This fractional derivative provides a fractional calculus parallel with the classical one, which applies to non-differentiable functions, and the present short article summarizes the main basic formulae so obtained.
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The first integral method for Wu---Zhang system with conformable time-fractional derivative
Mostafa Eslami,Hadi Rezazadeh +1 more
TL;DR: In this article, the first integral method was used to construct exact solutions of the Wu-Zhang system, which is based on the ring theory of commutative algebra, and the results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.
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Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio
TL;DR: Using the new fractional Taylor’s series, two new families of fractional Black–Scholes equations are derived, and some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics are made.
References
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Fractional Brownian Motions, Fractional Noises and Applications
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Stochastic Analysis of the Fractional Brownian Motion
TL;DR: In this article, the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations was proved for the Ito formula.
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A class of micropulses and antipersistent fractional brownian motion
TL;DR: In this article, the authors show that for up-and-down pulses with random moments of birth τ and random lifetime w determined by a Poisson random measure, when the pulse amplitude e → 0, while the pulse density δ increases to infinity, one obtains a process of fractal sum of micropulses.
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Stochastic differential equations with fractional Brownian motion input
TL;DR: In this article, the Liouville fractional derivative and the self-similarity property of fractional Brownian motion (FBM) were analyzed and the main statistical characteristics of FBM were derived.