Stochastic Pantograph Differential Equations Research Articles

Some important results for the conformable fractional stochastic pantograph differential equations in the L p space

Some important results for the conformable fractional stochastic pantograph differential equations in the L p space

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

Convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching

In this paper, we concern the strong convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching (SPDEs-MS). We present the balanced Euler method of SPDEs-MS and consider its moment boundedness under polynomial growth condition plus Khasminskii-type condition. We also study its strong convergence order and its mean-square stability. Two numerical examples are given to illustrate the theoretical results.

Khasminskii Approach for ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo Fractional Stochastic Pantograph Problem

In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained

Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps

Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps

Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps

The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.

Razumikhin Theorems on Polynomial Stability of Neutral Stochastic Pantograph Differential Equations with Markovian Switching

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work.

Theory of Stochastic Pantograph Differential Equations with θ-Caputo Fractional Derivative

Theory of Stochastic Pantograph Differential Equations with θ-Caputo Fractional Derivative

Stability of numerical solution to pantograph stochastic functional differential equations

The paper studies the convergence of the numerical solutions for pantograph stochastic functional differential equations which was proposed in Wu et al.(2022)[16]. We also show that the approximate solutions have the properties of almost surely polynomial stability and exponential stability.

Stability of neutral pantograph stochastic differential equations with generalized decay rate

In this paper, we investigate the stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs) with general decay rate. By applying the method of the Lyapunov function, the pth moment and almost sure stability with general decay rate of solution for NPSDEs are derived. Finally, an example is presented to show the effectiveness of the proposed methods.

Partial asymptotic stability of neutral pantograph stochastic differential equations with Markovian switching

In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.

Stability of hybrid pantograph stochastic functional differential equations

In this paper, we study a new type of stochastic functional differential equations which are called hybrid pantograph stochastic functional differential equations (HPSFDEs). We investigate moment properties and sample properties of the solutions such as the moment exponential stability, almost sure exponential stability and almost sure polynomial stability. Moreover, several theoretical examples and one practical example are given to illustrate our theory.

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

<p style='text-indent:20px;'>This paper focuses on the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, <inline-formula><tex-math id="M4">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.</p>

Analysis of stochastic pantograph differential equations with generalized derivative of arbitrary order

Abstract In this paper, we mainly study the existence of analytical solutions of stochastic pantograph differential equations. The standard Picard’s iteration method is used to obtain the theory.

Existence and Stability of Solutions for a Coupled System of Fuzzy Fractional Pantograph Stochastic Differential Equations

Existence and Stability of Solutions for a Coupled System of Fuzzy Fractional Pantograph Stochastic Differential Equations

Practical stability in relation to a part of variables of stochastic pantograph differential equations

The main purpose of this paper is to investigate the practical stability in relation to a part of the variables of stochastic pantograph differential equations (SPDEs, in short). With the aid of Lyapunov function method, the criterions of the global practical uniform asymptotic stability and the global practical uniform p-th moment exponential stability in relation to a part of variables of SPDEs are established. An illustrative example is given to show the effectiveness and feasibility of the proposed results on the practical stability with respect to a part of variables.

H-stability in pth moment of neutral pantograph stochastic differential equations with Markovian switching driven by Lévy noise

In this paper we investigate the h-stability in pth moment of neutral pantograph stochastic differential equations with Markovian switching driven by Lévy noise. The main tool used to prove the results is the Lyapunov method. We analyze two illustrative examples to show the interest and usefulness of the main results.

Formula omitted] th moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching

In this paper we focus on the pth moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEwMS). By means of the Lyapunov method, we develop some sufficient conditions on the pth moment exponential stability for NSPDEwMS. We analyze two examples to show the interest of the main results.

The [formula omitted]th moment asymptotical ultimate boundedness of pantograph stochastic differential equations with time-varying coefficients

This article examines the pth moment asymptotical ultimate boundedness of highly nonlinear pantograph stochastic differential equations. By Lyapunov function approach and stochastic analysis techniques, several novel criteria on the pth moment asymptotical boundedness are acquired. The proportional delays and different orders of nonlinearities have been considered and the diffusion operator is allowed to satisfy a weaker assumption including time-varying coefficients, which results in the improvements of the existing theory.

Asymptotical mean-square stability of linear θ-methods for stochastic pantograph differential equations: variable stepsize and transformation approach

The paper deals with the asymptotical mean-square stability of the linear θ-methods under variable stepsize and transformation approach for stochastic pantograph differential equations. A limiting equation for the analysis of numerical stability is introduced by Kronecker products. Under the condition which guarantee the stability of exact solutions, the optimal stability region of the linear θ-methods under variable stepsize is given by using the limiting equation, i.e. , which is the same to the deterministic problems. Moreover, the linear θ-methods under the transformation approach are also considered and the result of the stability is improved for . Finally, numerical examples are given to illustrate the asymptotical mean-square stability under variable stepsize and transformation approach.