Fractional Processes and Multidisciplinary Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (15 August 2023) | Viewed by 11067

Special Issue Editors


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Guest Editor
School of Management, Zhejiang University, Hangzhou 310058, China
Interests: data science and quantitative finance

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Guest Editor
School of Mathematics (Zhuhai), Sun Yat-sen University, Guangzhou 510275, China
Interests: statistical inference for stochastic processes; fractional Brownian motions; mixed fractional processes; filtering approach

Special Issue Information

Dear Colleagues,

Over the past few decades, numerous empirical studies have found that the phenomenon of long memory may be observed in the data of economics, hydrology, geophysics, climatology, telecommunication, crystallography, chemistry, and bioinformatics. Fractional processes, which display the memory property, have been widely used to describe natural and social phenomena. Some important fractional processes include fractional Brownian motions, sub-fractional Brownian motions, bi-fractional Brownian motions, multifractional Brownian motions and some other Gaussian processes. Since fractional processes are neither Markov processes nor semimartingales, the beautiful theories of stochastic analysis developed for semimartingale theory or for Markov processes cannot be applied. Some important mathematical tools to investigate stochastic integral for fractional processes are the Wick integral, the Stratonovich stochastic integral and the Young integral. Using these integrals, the statistical inference for stochastic differential equations driven by fractional processes has been the subject of active research for the last decade, besides being a challenging theoretical problem.

The aim of this Special Issue is to advance research on topics relating to the theory, implementation, and application of fractional processes. Potential topics include (but are not limited to):

  • Fractional/multifractional stochastic processes in economics, finance, hydrology, geophysics and climatology.
  • Self-similarity and (multi)scaling.
  • Mixed fractional processes.
  • Hurst exponent and Hölder regularity.
  • Rough volatility.
  • Fractional option prices.
  • Statistics inference of fractional models.

Prof. Dr. Weilin Xiao
Dr. Chunhao Cai
Guest Editors

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Keywords

  • fractional brownian motion
  • multifractional brownian motion
  • sub-fractional brownian motions
  • Bi-fractional brownian motions
  • rough volatility
  • fractional option prices
  • fractional models

Published Papers (9 papers)

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Research

28 pages, 9236 KiB  
Article
The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration
by Muhammad Bilal Riaz, Ali Raza Ansari, Adil Jhangeer, Muddassar Imran and Choon Kit Chan
Fractal Fract. 2023, 7(11), 809; https://doi.org/10.3390/fractalfract7110809 - 09 Nov 2023
Cited by 1 | Viewed by 944
Abstract
In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds [...] Read more.
In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds to stock returns. Employing the inverse scattering transformation, we find that the Cauchy problem for this model is insolvable. Consequently, we utilize the Φ6-expansion algorithm to generate generalized novel solitonic analytical wave structures within the system. We present graphical representations in contour, 3D, and 2D formats to illustrate how the system’s behavior responds to the propagation of pulses, enabling us to predict suitable parameter values that align with the data. Finally, a conclusion is given. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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11 pages, 283 KiB  
Article
Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion
by Ezzedine Mliki
Fractal Fract. 2023, 7(8), 591; https://doi.org/10.3390/fractalfract7080591 - 30 Jul 2023
Viewed by 830
Abstract
The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. [...] Read more.
The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
17 pages, 416 KiB  
Article
Parameter Estimation in Rough Bessel Model
by Yuliya Mishura and Anton Yurchenko-Tytarenko
Fractal Fract. 2023, 7(7), 508; https://doi.org/10.3390/fractalfract7070508 - 28 Jun 2023
Viewed by 544
Abstract
In this paper, we construct consistent statistical estimators of the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion with H<1/2. As an auxiliary result, we also prove the continuity of the [...] Read more.
In this paper, we construct consistent statistical estimators of the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion with H<1/2. As an auxiliary result, we also prove the continuity of the fractional Bessel process. The results are illustrated with simulations. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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18 pages, 928 KiB  
Article
Arbitrage in the Hermite Binomial Market
by Xuwen Cheng, Yiran Zheng and Xili Zhang
Fractal Fract. 2022, 6(12), 702; https://doi.org/10.3390/fractalfract6120702 - 27 Nov 2022
Cited by 1 | Viewed by 917
Abstract
Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite [...] Read more.
Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than 12. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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13 pages, 409 KiB  
Article
Controlled Parameter Estimation for The AR(1) Model with Stationary Gaussian Noise
by Lin Sun, Chunhao Cai and Min Zhang
Fractal Fract. 2022, 6(11), 643; https://doi.org/10.3390/fractalfract6110643 - 03 Nov 2022
Viewed by 946
Abstract
This paper deals with the maximum likelihood estimator for the parameter of first-order autoregressive models driven by the stationary Gaussian noises (Colored noise) together with an input. First, we will find the optimal input that maximizes the Fisher information, and then, with the [...] Read more.
This paper deals with the maximum likelihood estimator for the parameter of first-order autoregressive models driven by the stationary Gaussian noises (Colored noise) together with an input. First, we will find the optimal input that maximizes the Fisher information, and then, with the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the maximum likelihood estimator will be investigated. The results of the numerical simulation confirm the theoretical analysis and show that the proposed maximum likelihood estimator performs well in finite samples. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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27 pages, 22266 KiB  
Article
An Insight into the Impacts of Memory, Selling Price and Displayed Stock on a Retailer’s Decision in an Inventory Management Problem
by Mostafijur Rahaman, Reda M. S. Abdulaal, Omer A. Bafail, Manojit Das, Shariful Alam and Sankar Prasad Mondal
Fractal Fract. 2022, 6(9), 531; https://doi.org/10.3390/fractalfract6090531 - 19 Sep 2022
Cited by 15 | Viewed by 1986
Abstract
The present paper aims to demonstrate the combined impact of memory, selling price, and exhibited stock on a retailer’s decision to maximizing the profit. Exhibited stock endorses demand and low selling prices are also helpful for creating demand. The proposed mathematical model considers [...] Read more.
The present paper aims to demonstrate the combined impact of memory, selling price, and exhibited stock on a retailer’s decision to maximizing the profit. Exhibited stock endorses demand and low selling prices are also helpful for creating demand. The proposed mathematical model considers demand as a linear function of selling price and displayed inventory. This work utilized fractional calculus to design a memory-based decision-making environment. Following the analytical theory, an algorithm was designed, and by using the Mathematica software, we produced the numerical optimization results. Firstly, the work shows that memory negatively influences the retailer’s goal of maximum profit, which is the most important consequence of the numerical result. Secondly, raising the selling price will maximize the profit though the selling price, and demand will be negatively correlated. Finally, compared to the selling price, the influence of the visible stock is slightly lessened. The theoretical and numerical results ultimately imply that there can be no shortage and memory restrictions, leading to the highest average profit. The recommended approach may be used in retailing scenarios for small start-up businesses when a warehouse is required for continuous supply, but a showroom is not a top concern. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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25 pages, 358 KiB  
Article
Asymptotic Behavior on a Linear Self-Attracting Diffusion Driven by Fractional Brownian Motion
by Litan Yan, Xue Wu and Xiaoyu Xia
Fractal Fract. 2022, 6(8), 454; https://doi.org/10.3390/fractalfract6080454 - 20 Aug 2022
Cited by 1 | Viewed by 1088
Abstract
Let BH={BtH,t0} be a fractional Brownian motion with Hurst index 12H<1. In this paper, we consider the linear self-attracting diffusion: [...] Read more.
Let BH={BtH,t0} be a fractional Brownian motion with Hurst index 12H<1. In this paper, we consider the linear self-attracting diffusion: dXtH=dBtH+σXtHdtθ0tXsHXuHdsdt+νdt with X0H=0, where θ>0 and σ,νR are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, Math. Ann.303 (1995), 87–93). Our main aim is to study the large time behaviors. We show that the solution tσθHXtHXH converges in distribution to a normal random variable, as t tends to infinity, and obtain two strong laws of large numbers associated with the solution XH. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
21 pages, 456 KiB  
Article
Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays
by Boliang Lu, Quanxin Zhu and Ping He
Fractal Fract. 2022, 6(7), 385; https://doi.org/10.3390/fractalfract6070385 - 09 Jul 2022
Cited by 2 | Viewed by 1136
Abstract
This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then [...] Read more.
This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the q1th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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23 pages, 380 KiB  
Article
Stationary Wong–Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations
by Lauri Viitasaari and Caibin Zeng
Fractal Fract. 2022, 6(6), 303; https://doi.org/10.3390/fractalfract6060303 - 30 May 2022
Viewed by 1558
Abstract
In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of [...] Read more.
In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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