Recent Advances in Stochastic Differential Equations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (20 February 2023) | Viewed by 8849
Please contact the Guest Editor or the Journal Editor for any queries about the scope, discount, submission procedure and publication process.

Special Issue Editors

1. Faculty of Mathematics and Informatics, Sorbonne University, 75252 Paris, France
2. Department of Differential Equations, Sofia University, 1164 Sofia, Bulgaria
Interests: differential geometry; dynamic geometry; time scale calculus; dynamic equations on time scales; integral equations; ordinary differential equations; partial differential equations; stochastic differential equations; clifford algebras; clifford analysis; quaternion analysis; iso-mathematics
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass 51921, Saudi Arabia
Interests: stochastic partial differential equations; stochastic ordinary differential equations; time scale calculus; harmonic analysis; ordinary and partial differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Stochastic differential equations originated in the theory of Brownian motion in the work of Albert Einstein and Smoluchowski. The mathematical theory of stochastic differential equations was developed in 1940s through the work of Japanese mathematician Kiyosi Ito, who introduced the concept of stochastic calculus and initiated the study of stochastic differential equations. Stochastic differential equations are used to model various phenomena, such as unstable stock prices or physical systems subject to thermal fluctuations.

The Special Issue "Advances in Stochastic Differential Equations" is aimed at providing a platform for the publication of relevant research articles covering all aspects of stochastic differential equations, stochastic partial differential equations, the super-symmetric theory of stochastic differential equations, applications in disciplines such as as chaos, turbulence, self-organized criticality, probability theory as well as numerical methods for use in the solving of stochastic differential equations and stochastic partial differential equations.

Prof. Dr. Svetlin G. Georgiev
Dr. Khaled Zennir
Guest Editors

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Keywords

  • ito calculus
  • stochastic differential equation
  • existence of solutions
  • stock prices
  • brownian motion
  • chaos

Published Papers (6 papers)

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Research

19 pages, 326 KiB  
Article
Local Existence and Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source Term
Axioms 2023, 12(4), 343; https://doi.org/10.3390/axioms12040343 - 31 Mar 2023
Cited by 1 | Viewed by 918
Abstract
The aim of this paper is to investigate the local weak existence and vacuum isolating of solutions, asymptotic behavior, and blow-up of the solutions for a wave equation involving the fractional Laplacian with nonlinear source. By means of the Galerkin approximations, we prove [...] Read more.
The aim of this paper is to investigate the local weak existence and vacuum isolating of solutions, asymptotic behavior, and blow-up of the solutions for a wave equation involving the fractional Laplacian with nonlinear source. By means of the Galerkin approximations, we prove the local weak existence and finite time blow-up of the solutions and we give the upper and lower bounds for blow-up time. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
17 pages, 821 KiB  
Article
Stability for Weakly Coupled Wave Equations with a General Internal Control of Diffusive Type
Axioms 2023, 12(1), 48; https://doi.org/10.3390/axioms12010048 - 02 Jan 2023
Cited by 1 | Viewed by 947
Abstract
The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay [...] Read more.
The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
21 pages, 805 KiB  
Article
A New Topological Approach to Target the Existence of Solutions for Nonlinear Fractional Impulsive Wave Equations
Axioms 2022, 11(12), 721; https://doi.org/10.3390/axioms11120721 - 12 Dec 2022
Cited by 5 | Viewed by 936
Abstract
This paper considers a class of fractional impulsive wave equations and improves a previous results. In fact, this paper adopts a new topological approach to prove the existence of classical solutions with a complex arguments caused by impulsive perturbations. To the best of [...] Read more.
This paper considers a class of fractional impulsive wave equations and improves a previous results. In fact, this paper adopts a new topological approach to prove the existence of classical solutions with a complex arguments caused by impulsive perturbations. To the best of our knowledge, there is a severe lack of results related to such impulsive equations. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
13 pages, 598 KiB  
Article
Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations
Axioms 2022, 11(11), 658; https://doi.org/10.3390/axioms11110658 - 21 Nov 2022
Cited by 1 | Viewed by 1144
Abstract
In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, [...] Read more.
In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
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12 pages, 298 KiB  
Article
Backward Stochastic Differential Equations (BSDEs) Using Infinite-Dimensional Martingales with Subdifferential Operator
Axioms 2022, 11(10), 536; https://doi.org/10.3390/axioms11100536 - 08 Oct 2022
Cited by 2 | Viewed by 1040
Abstract
In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with subdifferential operators that are driven by infinite-dimensional martingales. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence and uniqueness of the [...] Read more.
In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with subdifferential operators that are driven by infinite-dimensional martingales. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence and uniqueness of the solution are established using Yosida approximations. Furthermore, as an application of the main result, we shall show that the backward stochastic partial differential equation driven by infinite-dimensional martingales with a continuous linear operator has a unique solution under the special condition that the Ft-progressively measurable generator F of the model we proposed in this paper equals zero. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
21 pages, 1634 KiB  
Article
Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response
Axioms 2021, 10(4), 323; https://doi.org/10.3390/axioms10040323 - 28 Nov 2021
Cited by 6 | Viewed by 1786
Abstract
Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of [...] Read more.
Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results. Full article
(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)
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