Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Luchko, Yuri

doi:10.3390/axioms5010006

Cited by 17 publications

(18 citation statements)

References 15 publications

(27 reference statements)

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“…To mix the advantages of both space and time fractionality, spacetime double-fractional diffusion has been introduced and extensively studied from the theoretical point of view [15,23,28,29,31,32,39]; however, it has only been recently considered in financial modeling [8,14,24,25,26]. It features a more complete structure than the simple composition of the time and space fractional models as it exhibits non trivial phenomena including larges jumps and memory effects, which can not be understood as a simple market time re-parametrization of an α-stable process.…”

Section: Introductionmentioning

confidence: 99%

Series representAtion of the Pricing Formula for the EuropeaN Option Driven by Space-Time Fractional Diffusion

Aguilar

Coste

Korbel

2018

FCAA

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In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. This series formula is obtained from the Mellin-Barnes representation of the option price with help of residue summation in C 2 . We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions. MSC 2010 : 26A33; 34A08; 91B25; 91G20

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Section: Introductionmentioning

confidence: 99%

Series representAtion of the Pricing Formula for the EuropeaN Option Driven by Space-Time Fractional Diffusion

Aguilar

Coste

Korbel

2018

FCAA

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“…i.e., in the case of the α-fractional diffusion equation that was analyzed in detail in [16][17][18]. Thus the α-fractional diffusion equation can be treated as a "right fractionalization" of the conventional diffusion equation.…”

Section: The Entropy Production Rates Of the Fractional Diffusion Promentioning

confidence: 99%

“…The n-dimensional case of this equation was analyzed in Reference [7]. In Reference [16], it was shown that the entropy production rate of the fundamental solution to the one-dimensional α-fractional diffusion equation is exactly the same as in the case of the conventional diffusion equation. The α-fractional diffusion equation is a PDE with the Caputo time-fractional derivative of order α, 0 < α ≤ 1 and the fractional spatial derivative of order 2α.…”

Section: Introductionmentioning

confidence: 99%

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Luchko

2019

Entropy

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Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α ≤ 2 . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension n ∈ N if 0 < β ≤ 1 , 0 < α ≤ 2 and additionally for 1 < β ≤ α ≤ 2 in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders β and α of the time and spatial derivatives and on the space dimension n and is given by the expression β n α t , t being the time variable. Even if it is an increasing function in β , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition 0 < β ≤ 1 , i.e., only the slow and the conventional diffusion can be described by this equation.

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“…τ contains a Gammafunction with the same parameter, namely Γ 1 − s 2 , that could be canceled in their product. This feature is valid only in the two-dimensional case, whereas the Mellin-Barnes integrals of the type (3.14) are essentially more complicated both in the one-and in the three-dimensional cases (see the very recent paper [12] for the results in the one-dimensional case).…”

Section: Fundamental Solution Of the α-Fractional Diffusion Equationmentioning

confidence: 99%

“…Let us mention that the α-fractional diffusion equation is a particular case of the more general time-and space-fractional diffusion equation that has been analyzed in the one-dimensional case in [21] and in the multi-dimensional case in [6] and [29] to mention only few of many relevant publications. In [12], the case of the one-dimensional α-fractional diffusion equation was considered in detail.…”

Section: Introductionmentioning

confidence: 99%

A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis

Luchko

2016

Math. Model. Nat. Phenom.

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In this paper, a special model for the two-dimensional anomalous diffusion is first deduced from the basic continuous time random walk equations in terms of a time-and spacefractional partial differential equation with the Caputo time-fractional derivative of order α/2 and the Riesz space-fractional derivative of order α. For α < 2, this α-fractional diffusion equation describes the so called Lévy flights that correspond to the continuous time random walk model, where both the mean waiting time and the jump length variance of the diffusing particles are divergent. The fundamental solution to the α-fractional diffusion equation is shown to be a two-dimensional probability density function that can be expressed in explicit form in terms of the Mittag-Leffler function depending on the auxiliary variable |x|/(2 √ t) as in the case of the fundamental solution to the classical isotropic diffusion equation. Moreover, we show that the entropy production rate associated with the anomalous diffusion process described by the α-fractional diffusion equation is exactly the same as in the case of the classical isotropic diffusion equation. Thus the α-fractional diffusion equation can be considered to be a natural generalization of the classical isotropic diffusion equation that exhibits some characteristics of both anomalous and classical diffusion.

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Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

Cited by 17 publications

References 15 publications

Series representAtion of the Pricing Formula for the EuropeaN Option Driven by Space-Time Fractional Diffusion

Series representAtion of the Pricing Formula for the EuropeaN Option Driven by Space-Time Fractional Diffusion

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis

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