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Mathematics
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Advances in Stochastic Differential Equations and Applications
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Advances in Stochastic Differential Equations and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 31 July 2026 | Viewed by 2794

Special Issue Editor

Prof. Dr. Yonggui Kao

E-Mail Website
Guest Editor
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China
Interests: stochastic differential equations and applications; stochastic system control; variable structure control theory and application; complex system and complex network theory; distributed parameter system theory and application; fractional order differential equations and applications; unmanned system control

Special Issue Information

Dear Colleagues,

Stochastic differential equations (SDEs) serve as a fundamental tool in modelling dynamical systems influenced by random effects, with wide-ranging applications in science, engineering, finance, and beyond. This Special Issue aims to highlight recent advances in the theory, analysis, and application of stochastic differential equations and related mathematical frameworks.

We invite high-quality original research papers and comprehensive reviews on topics including, but not limited to, the following:

  • Theoretical developments and numerical methods for stochastic differential equations and impulse differential equations;
  • Control theory for stochastic systems, including stochastic system control and variable structure control;
  • Applications and theory of complex systems and complex network models influenced by stochastic dynamics;
  • Distributed parameter system theory and its applications in stochastic contexts;
  • Fractional order differential equations and their stochastic counterparts;
  • Control and modeling of unmanned systems under stochastic influences.

This Special Issue provides a platform to explore novel methodologies, interdisciplinary approaches, and practical applications that deepen the understanding and extend the applicability of stochastic differential equations and their generalizations.

Prof. Dr. Yonggui Kao
Guest Editor

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 250 words) can be sent to the Editorial Office for assessment.

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • stochastic differential equations
  • stochastic system control
  • variable structure control theory
  • complex theory
  • complex network theory
  • distributed parameter system theory

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

Published Papers (5 papers)

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Research

25 pages, 1848 KB  
Article
Research on American Option Pricing Under the Heston Jump Diffusion Model—Based on Fourier Space Time-Stepping Method
by Yu Zhang, Shilong Wang and Longsuo Li
Mathematics 2026, 14(9), 1412; https://doi.org/10.3390/math14091412 - 23 Apr 2026
Viewed by 236
Abstract
American options are more complex to price than European options because they grant holders the right to exercise at any time before expiration, especially in realistic market environments that consider both stochastic volatility and asset price jumps. Therefore, this paper studies the pricing [...] Read more.
American options are more complex to price than European options because they grant holders the right to exercise at any time before expiration, especially in realistic market environments that consider both stochastic volatility and asset price jumps. Therefore, this paper studies the pricing of American options under the Heston stochastic volatility model, incorporating the Merton jump-diffusion process. For this high-dimensional, nonlinear free boundary problem, this paper adopts the Fourier space time-stepping method for numerical solution. This method utilizes the characteristic function in Fourier space to implement time-stepping, effectively addressing computational difficulties caused by stochastic volatility and jump processes, and it determines the optimal exercise boundary by comparing the holding value with the immediate exercise value at each step. Numerical experiments show that the method is computationally stable and accurate, clearly capturing the early exercise premium and dynamic changes in the exercise boundary. Additionally, parameter sensitivity analysis reveals that the jump component significantly affects option value (with a premium of approximately 6.74%), highlighting the necessity of incorporating jump risk into pricing models. This work provides an effective numerical framework for American option pricing under stochastic volatility and jump environments, possessing both theoretical significance and practical application value. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications)
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16 pages, 534 KB  
Article
A Stochastic Model Predictive Control Strategy for Vehicle Routing with Correlated Stochastic Service Times
by Guosong He, Qiuchi Li, Xingchen Li, Yu Huang, Yi Huang and Qianqian Duan
Mathematics 2026, 14(6), 1032; https://doi.org/10.3390/math14061032 - 18 Mar 2026
Viewed by 325
Abstract
Uncertainty in travel and service times poses significant challenges for vehicle routing in logistics systems. This paper proposes a stochastic model predictive control (SMPC) strategy to manage a Vehicle Routing Problem with time windows (VRPTW) under stochastic service times with correlation across customers. [...] Read more.
Uncertainty in travel and service times poses significant challenges for vehicle routing in logistics systems. This paper proposes a stochastic model predictive control (SMPC) strategy to manage a Vehicle Routing Problem with time windows (VRPTW) under stochastic service times with correlation across customers. The approach combines a dynamic optimization model with single and joint chance constraints and a forecasting tool for updating travel plans as new information becomes available. A deterministic reformulation of the stochastic constraints is developed so that the problem can be solved via mixed-integer programming. The aim of this paper is to demonstrate that the SMPC strategy can maintain a high level of time-window reliability (meeting customer time windows with high probability) at a reasonable cost by re-optimizing routes over a moving horizon. In numerical case studies, the SMPC approach achieves the desired reliability levels while incurring only modest increases in total cost, and it flexibly adjusts the cost–risk tradeoff by switching between single and joint chance constraints. These results illustrate the potential of the proposed method for real-time distribution routing under uncertainty and highlight the novel contribution of integrating chance-constrained optimization with Model Predictive Control in a VRPTW context. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications)
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30 pages, 755 KB  
Article
Adaptive Fault-Tolerant Sliding Mode Control for Itô-Type Stochastic Time-Delay Markov Jump Systems with Partly Unknown Transition Probabilities
by Tengyu Ma, Minli Zheng, Lijun Zhang and Longsuo Li
Mathematics 2026, 14(6), 1001; https://doi.org/10.3390/math14061001 - 16 Mar 2026
Viewed by 354
Abstract
This study addresses the challenge of designing an adaptive sliding mode controller for a class of nonlinear Markov jump systems. These systems are characterized by unmeasurable states, partially unknown transition probabilities, and uncertainties arising from matched external disturbances and modeling inaccuracies. In control [...] Read more.
This study addresses the challenge of designing an adaptive sliding mode controller for a class of nonlinear Markov jump systems. These systems are characterized by unmeasurable states, partially unknown transition probabilities, and uncertainties arising from matched external disturbances and modeling inaccuracies. In control design and analysis, the nonlinear Markov system in which both the linear term and specific information about the upper bound in the external disturbance term are unknown. To enable descending equivalent sliding mode motion to regulate the dithering phenomenon in a controlled system, an integral sliding surface is established to achieve chattering suppression via descending equivalent sliding motion. A key theoretical contribution is the rigorous proof that the proposed control law ensures both finite-time reachability of the sliding surface and mean-square stability of the closed-loop trajectories. Comparative simulation results demonstrate that the proposed approach achieves a state estimation RMSE of 0.175, which is 48.0% lower than conventional sliding mode control (0.337) and 3.3% lower than observer-based sliding mode control without fault compensation (0.181). The controller reduces control chattering by 75.2% compared to conventional SMC (total variation from 64.4 to 16.0), achieves sliding surface reachability within 0.42s, and maintains effective fault estimation with an average RMSE of 0.138 for time-varying actuator efficiency factors. These quantitative improvements validate the effectiveness of the proposed fault-tolerant mechanism. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications)
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10 pages, 406 KB  
Article
Numerical Algorithms for Acoustic Wave Propagation in Pipelines via a Class of Stochastic Partial Differential Systems
by Xinrong Cong, Longsuo Li and Shuxia Zhang
Mathematics 2026, 14(1), 86; https://doi.org/10.3390/math14010086 - 26 Dec 2025
Viewed by 455
Abstract
A class of partial differential equations with random noise is employed to model the pipe acoustic system. A high-precision compact differential scheme is constructed for its solution. To ensure numerical stability, a buffer layer technique is applied to absorb outgoing waves. The propagation [...] Read more.
A class of partial differential equations with random noise is employed to model the pipe acoustic system. A high-precision compact differential scheme is constructed for its solution. To ensure numerical stability, a buffer layer technique is applied to absorb outgoing waves. The propagation of acoustic waves under different modes is simulated. Furthermore, a specific numerical example is provided, and the results show good agreement with theoretical analysis. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications)
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17 pages, 1307 KB  
Article
Exponentially Fitted Midpoint Scheme for a Stochastic Oscillator
by Yaru Wang, Zhenyu Lang, Xiuling Yin and Zihan Zhao
Mathematics 2026, 14(1), 17; https://doi.org/10.3390/math14010017 - 21 Dec 2025
Viewed by 423
Abstract
In this paper, we propose the exponentially fitted midpoint scheme for the stochastic oscillator. This scheme is first-order strongly convergent and it preserves symplectic. It can effectively simulate the oscillatory behavior of stochastic oscillators, and its second moment grows linearly with time. In [...] Read more.
In this paper, we propose the exponentially fitted midpoint scheme for the stochastic oscillator. This scheme is first-order strongly convergent and it preserves symplectic. It can effectively simulate the oscillatory behavior of stochastic oscillators, and its second moment grows linearly with time. In addition, we also propose a two-parameter estimation method by analyzing the expectation and variance in the discrete scheme. Numerical experiments are given to verify effectiveness of the exponential fitting method and parameter estimation methods based on this scheme. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications)
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