Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio

Jumarie, Guy

doi:10.1016/j.camwa.2009.05.015

Cited by 238 publications

(95 citation statements)

References 21 publications

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“…One can refer to Podlubny (1999) for a survey of applications of fractional calculus. Ezzat (2010Ezzat ( , 2011a was the first writer who established a new formula of heat conduction law by using the new Taylor-Riemann series expansion of time-fractional order a developed by Jumarie (2010) as follows:…”

Section: Introductionmentioning

confidence: 99%

Fractional thermoelasticity applications for porous asphaltic materials

Ezzat

2016

Pet. Sci.

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A new mathematical model of poro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with fractional order. One-dimensional application for a poroelastic half-space saturated with fluid is considered. The surface of the halfspace is assumed to be traction-free, permeable, and subjected to heating. The Laplace transform technique is used to solve the problem. The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature, stresses, strains, and displacements will be illustrated graphically for the solid and the liquid.

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Section: Introductionmentioning

confidence: 99%

Fractional thermoelasticity applications for porous asphaltic materials

Ezzat

2016

Pet. Sci.

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“…A function continuous everywhere, but nowhere integerdifferentiable, necessarily exhibits random-like or pseudo-random features, in that various samplings of these functions, in the same given interval, will be different. This may explain the huge amount of literature extending the theory of stochastic differential equations to describe stochastic dynamics driven by fractional Brownian motion [25,33,34]. Regarding the anomalous properties of space-time with multifractal structure, we highlight the interesting work in ref.…”

Section: Introductionmentioning

confidence: 99%

On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

Weberszpil

Lazo²,

Helayël-Neto

2015

Physica A: Statistical Mechanics and its Applications

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Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the q-calculus in Tsallis' version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems; Kaniadakis' approach, based on the compatibility between relativity and thermodynamics;Fractional Calculus (FC), that deals with the dynamics of anomalous transport and other natural phenomena, and also some local versions of FC that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations.The question we might ask is whether or not there are common aspects that connect these alternative approaches.In this short communication, we discuss a possible relationship between q−deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators defined in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. This connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics, such as porous or granular media. Possible connections with other alternative definitions of FC are also contempled. Insights on complexity connected to concepts like coarse-grained space-time and physics in general are pointed out.

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“…In 1996, Heyde and Dai researched the issues related to the fractional Brownian motion, and introduced the relationship with short-term dependence (SRD) and long-range dependence (LRD), and got the SDE definition driven by the fractional Brownian motion [30][31][32][33]. In 2002, Necula simulated the change process of asset returns based on the SDE driven by FBM.…”

Section: Stochastic Differential Equations (Sde) Based On Fbmmentioning

confidence: 99%

Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution

Song

Liang

2016

Entropy

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Abstract:In this paper, we propose a novel framework for the diagnosis of incipient bearing faults and trend prediction of weak faults which result in gradual aggravation with the bearing vibration intensity as the characteristic parameter. For the weak fault diagnosis, the proposed framework adopts the improved minimum entropy deconvolution (MED) theory to identify the weak fault characteristics of mechanical equipment. From a large number of actual data analysis, once a bearing shows a weak fault, the bearing vibration intensity not only has random non-stationary, but also long-range dependent (LRD) characteristics. Therefore, the stochastic model with LRD−fractional Brown motion (FBM) is proposed to evaluate and predict the condition of slowly varying bearing faults which is a gradual process from weak fault occurrence to severity. For the FBM stochastic model, we mainly implement the derivation and the parameter identification of the FBM model. This is the first study to slowly fault prediction with stochastic model FBM. Experimental results show that the proposed methods can obtain the best performance in incipient fault diagnosis and bearing condition trend prediction.

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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

Cited by 238 publications

References 21 publications

Fractional thermoelasticity applications for porous asphaltic materials

Fractional thermoelasticity applications for porous asphaltic materials

On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution

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