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26 pages, 1066 KiB

Open AccessArticle

Fractional Gaussian Noise: Projections, Prediction, Norms

by Iryna Bodnarchuk, Yuliya Mishura and Kostiantyn Ralchenko

Fractal Fract. 2025, 9(7), 428; https://doi.org/10.3390/fractalfract9070428 - 29 Jun 2025

Viewed by 248

We examine the one-sided and two-sided (bilateral) projections of an element of fractional Gaussian noise onto its neighboring elements. We establish several analytical results and conduct a numerical study to analyze the behavior of the coefficients of these projections as functions of the [...] Read more.

We examine the one-sided and two-sided (bilateral) projections of an element of fractional Gaussian noise onto its neighboring elements. We establish several analytical results and conduct a numerical study to analyze the behavior of the coefficients of these projections as functions of the Hurst index and the number of neighboring elements used for the projection. We derive recurrence relations for the coefficients of the two-sided projection. Additionally, we explore the norms of both types of projections. Certain special cases are investigated in greater detail, both theoretically and numerically. Full article

(This article belongs to the Special Issue Stochastic Models, Fractional Calculus and Non-Local Operators: Theoretical Results and Applications)

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15 pages, 323 KiB

Open AccessArticle

Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity

by Xinguang Zhang, Jingsong Chen, Peng Chen, Lishuang Li and Yonghong Wu

Fractal Fract. 2024, 8(6), 337; https://doi.org/10.3390/fractalfract8060337 - 5 Jun 2024

Viewed by 1197

Abstract

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to [...] Read more.

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

11 pages, 312 KiB

Open AccessArticle

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

by Zhidong Guo, Yang Liu and Linsong Dai

Fractal Fract. 2024, 8(1), 13; https://doi.org/10.3390/fractalfract8010013 - 22 Dec 2023

Cited by 3 | Viewed by 2104

Abstract

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are [...] Read more.

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments. Full article

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30 pages, 692 KiB

Open AccessArticle

Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes

by Anatoliy Malyarenko, Yuliya Mishura, Kostiantyn Ralchenko and Yevheniia Anastasiia Rudyk

Axioms 2023, 12(11), 1026; https://doi.org/10.3390/axioms12111026 - 31 Oct 2023

Cited by 6 | Viewed by 1956

Abstract

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian [...] Read more.

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes. Full article

(This article belongs to the Special Issue Stochastic Processes in Quantum Mechanics and Classical Physics)

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15 pages, 2404 KiB

Open AccessArticle

Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment

by Binxin Ji, Xiangxing Tao and Yanting Ji

Fractal Fract. 2022, 6(5), 244; https://doi.org/10.3390/fractalfract6050244 - 29 Apr 2022

Cited by 11 | Viewed by 2497

Abstract

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding

I t \hat{o}

’s formula, the B-S type PDE is derived by a self-financing strategy. [...] Read more.

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding

I t \hat{o}

’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option, respectively. Full article

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19 pages, 404 KiB

Open AccessReview

Estimating Drift Parameters in a Sub-Fractional Vasicek-Type Process

by Anas D. Khalaf, Tareq Saeed, Reman Abu-Shanab, Waleed Almutiry and Mahmoud Abouagwa

Entropy 2022, 24(5), 594; https://doi.org/10.3390/e24050594 - 24 Apr 2022

Cited by 5 | Viewed by 2094

Abstract

This study deals with drift parameters estimation problems in the sub-fractional Vasicek process given by

d x_{t} = θ (μ - x_{t}) d t + d S_{t}^{H}

, with

θ > 0

,

μ \in R

being [...] Read more.

This study deals with drift parameters estimation problems in the sub-fractional Vasicek process given by

d x_{t} = θ (μ - x_{t}) d t + d S_{t}^{H}

, with

θ > 0

,

μ \in R

being unknown and

t \geq 0

; here,

S^{H}

represents a sub-fractional Brownian motion (sfBm). We introduce new estimators

\hat{θ}

for

θ

and

\hat{μ}

for

μ

based on discrete time observations and use techniques from Nordin–Peccati analysis. For the proposed estimators

\hat{θ}

and

\hat{μ}

, strong consistency and the asymptotic normality were established by employing the properties of

S^{H}

. Moreover, we provide numerical simulations for sfBm and related Vasicek-type process with different values of the Hurst index H. Full article

(This article belongs to the Section Entropy Reviews)

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13 pages, 294 KiB

Open AccessArticle

Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

by Bodo Herzog

Mathematics 2022, 10(3), 340; https://doi.org/10.3390/math10030340 - 23 Jan 2022

Cited by 3 | Viewed by 3607

Abstract

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions

{B_{t}^{H}, t \geq 0}

[...] Read more.

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions

{B_{t}^{H}, t \geq 0}

and sub-fractional Brownian motions

{ξ_{t}^{H}, t \geq 0}

with Hurst parameter

H \in (\frac{1}{2}, 1)

. We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

17 pages, 339 KiB

Open AccessArticle

Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

by Chun-Hao Cai, Jun-Qi Hu and Ying-Li Wang

Fractal Fract. 2021, 5(4), 226; https://doi.org/10.3390/fractalfract5040226 - 17 Nov 2021

Cited by 3 | Viewed by 1688

Abstract

In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates [...] Read more.

In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates for the small

L^{2}

-ball probabilities for a sub-fractional Brownian motion are derived. Asymptotic analysis on the Karhunen–Loève eigenvalues for the corresponding “derivative” process is also established. Full article

(This article belongs to the Special Issue Stochastic Calculus for Fractional Brownian Motion)

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17 pages, 2316 KiB

Open AccessArticle

Reducing Myosin II and ATP-Dependent Mechanical Activity Increases Order and Stability of Intracellular Organelles

by Ishay Wohl and Eilon Sherman

Int. J. Mol. Sci. 2021, 22(19), 10369; https://doi.org/10.3390/ijms221910369 - 26 Sep 2021

Cited by 3 | Viewed by 2743

Abstract

Organization of intracellular content is affected by multiple simultaneous processes, including diffusion in a viscoelastic and structured environment, intracellular mechanical work and vibrations. The combined effects of these processes on intracellular organization are complex and remain poorly understood. Here, we studied the organization [...] Read more.

Organization of intracellular content is affected by multiple simultaneous processes, including diffusion in a viscoelastic and structured environment, intracellular mechanical work and vibrations. The combined effects of these processes on intracellular organization are complex and remain poorly understood. Here, we studied the organization and dynamics of a free Ca⁺⁺ probe as a small and mobile tracer in live T cells. Ca⁺⁺, highlighted by Fluo-4, is localized in intracellular organelles. Inhibiting intracellular mechanical work by myosin II through blebbistatin treatment increased cellular dis-homogeneity of Ca⁺⁺-rich features in length scale < 1.1 μm. We detected a similar effect in cells imaged by label-free bright-field (BF) microscopy, in mitochondria-highlighted cells and in ATP-depleted cells. Blebbistatin treatment also reduced the dynamics of the Ca⁺⁺-rich features and generated prominent negative temporal correlations in their signals. Following Guggenberger et al. and numerical simulations, we suggest that diffusion in the viscoelastic and confined medium of intracellular organelles may promote spatial dis-homogeneity and stability of their content. This may be revealed only after inhibiting intracellular mechanical work and related cell vibrations. Our described mechanisms may allow the cell to control its organization via balancing its viscoelasticity and mechanical activity, with implications to cell physiology in health and disease. Full article

(This article belongs to the Collection Feature Papers in Molecular Biophysics)

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17 pages, 340 KiB

Open AccessArticle

Lévy Processes Linked to the Lower-Incomplete Gamma Function

by Luisa Beghin and Costantino Ricciuti

Fractal Fract. 2021, 5(3), 72; https://doi.org/10.3390/fractalfract5030072 - 17 Jul 2021

Cited by 4 | Viewed by 2142

Abstract

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to [...] Read more.

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. Full article

(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)

29 pages, 2358 KiB

Open AccessArticle

Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation

by Richard L. Magin and Ervin K. Lenzi

Mathematics 2021, 9(13), 1481; https://doi.org/10.3390/math9131481 - 24 Jun 2021

Cited by 7 | Viewed by 2416

Abstract

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct [...] Read more.

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions. Full article

(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)

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16 pages, 2953 KiB

Open AccessArticle

Fractional Order Complexity Model of the Diffusion Signal Decay in MRI

by Richard L. Magin, Hamid Karani, Shuhong Wang and Yingjie Liang

Mathematics 2019, 7(4), 348; https://doi.org/10.3390/math7040348 - 12 Apr 2019

Cited by 26 | Viewed by 5053

Abstract

Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can [...] Read more.

Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease. Full article

(This article belongs to the Special Issue Fractional Order Systems)

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17 pages, 828 KiB

Open AccessArticle

Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion

by Shengfeng Li and Yi Dong

Algorithms 2018, 11(12), 197; https://doi.org/10.3390/a11120197 - 4 Dec 2018

Cited by 6 | Viewed by 3616

Abstract

In the paper, we tackle the least squares estimators of the Vasicek-type model driven by sub-fractional Brownian motion:

d X_{t} = (μ + θ X_{t}) d t + d S_{t}^{H}, t \geq 0

with [...] Read more.

In the paper, we tackle the least squares estimators of the Vasicek-type model driven by sub-fractional Brownian motion:

d X_{t} = (μ + θ X_{t}) d t + d S_{t}^{H}, t \geq 0

with

X_{0} = 0

, where

S^{H}

is a sub-fractional Brownian motion whose Hurst index H is greater than

\frac{1}{2}

, and

μ \in R

,

θ \in R^{+}

are two unknown parameters. Based on the so-called continuous observations, we suggest the least square estimators of

μ

and

θ

and discuss the consistency and asymptotic distributions of the two estimators. Full article

(This article belongs to the Special Issue Parameter Estimation Algorithms and Its Applications)

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