Fractal and Fractional

Research

28 pages, 9236 KiB

Open AccessArticle

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 - 9 Nov 2023

Cited by 3 | Viewed by 1372

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

Φ^{6}

-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

Open AccessArticle

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 1137

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

Open AccessArticle

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 771

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

Open AccessArticle

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 1264

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

\frac{1}{2}

. 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

Open AccessArticle

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 - 3 Nov 2022

Viewed by 1291

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

Open AccessArticle

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 18 | Viewed by 2695

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

Open AccessArticle

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 1425

Abstract

Let

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

be a fractional Brownian motion with Hurst index

\frac{1}{2} \leq H < 1

. In this paper, we consider the linear self-attracting diffusion: [...] Read more.

Let

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

be a fractional Brownian motion with Hurst index

\frac{1}{2} \leq H < 1

. In this paper, we consider the linear self-attracting diffusion:

d X_{t}^{H} = d B_{t}^{H} + σ X_{t}^{H} d t - θ (\int_{0}^{t} (X_{s}^{H} - X_{u}^{H}) d s) d t

+ ν d t

with

X_{0}^{H} = 0

, where

θ > 0

and

σ, ν \in R

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

{(t - \frac{σ}{θ})}^{H} (X_{t}^{H} - X_{\infty}^{H})

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

X^{H}

. Full article

(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)

21 pages, 456 KiB

Open AccessArticle

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 - 9 Jul 2022

Cited by 3 | Viewed by 1482

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

q_{1}

th 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

Open AccessArticle

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 1964

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

L^{p} (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

L^{p} (Ω)

. Our stationary approximation is suitable for all values of

H \in (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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