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Fractal Fract
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Analysis of Fractional Stochastic Differential Equations and Their Applications
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Analysis of Fractional Stochastic Differential Equations and Their Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 7776

Special Issue Editors

Prof. Dr. Zhiguo Luo

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
Interests: fractional differential equations; stochastic differential equations; stability analysis; impulsive differential equations; difference equations and their applications
Dr. Danfeng Luo

E-Mail Website
Guest Editor
Department of Mathematics, Guizhou University, Guiyang 550025, China
Interests: averaging principle in stochastic systems; stability or controllability in fractional differential equations; fuzzy differential equations

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to communicate and collect results on fractional stochastic differential equations and their applications. We invite submissions of high-quality articles on the existence, uniqueness, stability, controllability and averaging principle of solutions. This Special Issue, “Analysis of Fractional Stochastic Differential Equations and Their Applications”, focuses on a wide range of topics in fractional stochastic analysis and its applications, including, but not limited to, the following:

  • Finite-time stability
  • Ulam–Hyers stability
  • Controllability
  • Averaging principle
  • Existence or uniqueness
  • Delay differential equations
  • Impulsive differential equations
  • Fuzzy differential equations

Prof. Dr. Zhiguo Luo
Dr. Danfeng Luo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional differential equations
  • stochastic differential equations
  • delay differential equations
  • impulsive differential equations
  • fuzzy differential equations
  • stability analysis
  • averaging principle
  • controllability
  • averaging principle
  • existence or uniqueness

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (8 papers)

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Research

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27 pages, 392 KiB  
Article
L1 Scheme for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise
by Xiaolei Wu and Yubin Yan
Fractal Fract. 2025, 9(3), 173; https://doi.org/10.3390/fractalfract9030173 - 12 Mar 2025
Viewed by 454
Abstract
This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter H∈(1/2,1). The finite element method is employed for spatial [...] Read more.
This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter H∈(1/2,1). The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order α∈(0,1) and the Riemann–Liouville time-fractional integral of order γ∈(0,1), respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order O(τmin{H+α+γ−1−ε,α}),ε>0. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
24 pages, 362 KiB  
Article
Stability and Controllability Analysis of Stochastic Fractional Differential Equations Under Integral Boundary Conditions Driven by Rosenblatt Process with Impulses
by Mohamed S. Algolam, Sadam Hussain, Bakri A. I. Younis, Osman Osman, Blgys Muflh, Khaled Aldwoah and Nidal Eljaneid
Fractal Fract. 2025, 9(3), 146; https://doi.org/10.3390/fractalfract9030146 - 26 Feb 2025
Cited by 1 | Viewed by 717
Abstract
Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the [...] Read more.
Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive ψ-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
14 pages, 340 KiB  
Article
Successive Approximation and Stability Analysis of Fractional Stochastic Differential Systems with Non-Gaussian Process and Poisson Jumps
by Nidhi Asthana, Mohd Nadeem and Rajesh Dhayal
Fractal Fract. 2025, 9(2), 130; https://doi.org/10.3390/fractalfract9020130 - 19 Feb 2025
Viewed by 410
Abstract
This paper investigates a new class of fractional stochastic differential systems with non-Gaussian processes and Poisson jumps. Firstly, we examine the solvability results for the considered system. Furthermore, new stability results for the proposed system are derived. The findings are established through the [...] Read more.
This paper investigates a new class of fractional stochastic differential systems with non-Gaussian processes and Poisson jumps. Firstly, we examine the solvability results for the considered system. Furthermore, new stability results for the proposed system are derived. The findings are established through the application of Grönwall’s inequality, the successive approximation method, and the corollary of the Bihari inequality. Finally, the validity of the results is proved through an example. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
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17 pages, 410 KiB  
Article
Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2
by Anurag Shukla, Sumati Kumari Panda, Velusamy Vijayakumar, Kamalendra Kumar and Kothandabani Thilagavathi
Fractal Fract. 2024, 8(9), 499; https://doi.org/10.3390/fractalfract8090499 - 24 Aug 2024
Cited by 6 | Viewed by 1082
Abstract
This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1<q<2. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects [...] Read more.
This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1<q<2. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
18 pages, 5746 KiB  
Article
Remaining Useful Life Prediction for Power Storage Electronic Components Based on Fractional Weibull Process and Shock Poisson Model
by Wanqing Song, Xianhua Yang, Wujin Deng, Piercarlo Cattani and Francesco Villecco
Fractal Fract. 2024, 8(8), 485; https://doi.org/10.3390/fractalfract8080485 - 19 Aug 2024
Cited by 1 | Viewed by 1271
Abstract
For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound [...] Read more.
For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound Poisson process (NHCP) is proposed to simulate the shock effect in the DTS model. Considering the long-range dependence and heavy-tailed characteristics of the degradation process, fractional Weibull process (fWp) is employed in the diffusion term of the stochastic degradation model. Furthermore, the drift and diffusion coefficients are constantly updated to describe the environmental interference. Prior to the model training, steady degradation and shock data must be separated, based on the three-sigma principle. Degradation data for the lithium-ion batteries (LIBs) and ultracapacitors are employed for model verification under different operation protocols in the power system. Recent deep learning models and stochastic process-based methods are utilized for model comparison, and the proposed model shows higher prediction accuracy. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
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15 pages, 315 KiB  
Article
Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process
by Ghada AlNemer, Mohamed Hosny, Ramalingam Udhayakumar and Ahmed M. Elshenhab
Fractal Fract. 2024, 8(6), 342; https://doi.org/10.3390/fractalfract8060342 - 6 Jun 2024
Cited by 1 | Viewed by 1013
Abstract
Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, [...] Read more.
Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
23 pages, 640 KiB  
Article
A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion
by Marc Mukendi Mpanda
Fractal Fract. 2024, 8(6), 330; https://doi.org/10.3390/fractalfract8060330 - 30 May 2024
Viewed by 1094
Abstract
This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven [...] Read more.
This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter H∈(0,1). We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
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Other

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9 pages, 340 KiB  
Brief Report
Modeling Double Stochastic Opinion Dynamics with Fractional Inflow of New Opinions
by Vygintas Gontis
Fractal Fract. 2024, 8(9), 513; https://doi.org/10.3390/fractalfract8090513 - 29 Aug 2024
Viewed by 783
Abstract
Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second [...] Read more.
Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second form of the Pareto distribution. We elucidate this distinctive power-law statistical property through the lens of agent heterogeneity in trading activity and asset possession. Our study introduces a novel modeling approach that combines fractional Lévy stable motion for limit order inflow with this power-law distribution for cancellation times, significantly enhancing the prediction of order imbalances. This model not only addresses gaps in current financial market modeling but also extends to broader contexts such as opinion dynamics in social systems, capturing the finite lifespan of opinions. Characterized by stationary increments and a departure from self-similarity, our model provides a unique framework for exploring long-range dependencies in time series. This work paves the way for more precise financial market analyses and offers new insights into the dynamic nature of opinion formation in social systems. Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
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