Neutral Stochastic Functional Differential Equations Research Articles

On the existence and mean square stability of neutral stochastic functional differential equations driven by G‐Lévy process

The goal of the present article is to investigate the concepts of existences and stability for neutral stochastic functional differential equations (NSFDEs) driven by G‐Lévy process. Under non‐Lipschitz assumption the existence and uniqueness result for solutions of NSFDEs based on G‐Lévy process has been acquired. The results have been deducted by using the Bihari inequality, H lder inequality and the Burkholder–Davis–Gundy (BDG) type inequalities of the G‐framework. In addition, the mean square stability has been studied.

Comparison theorem for neutral stochastic functional differential equations driven by [formula omitted]-Brownian motion

In this paper, we investigate sufficient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case (Bai and Jiang, 2016) and nonlinear expectation framework (Huang and Yang, 2021).

Ulam–Hyers–Rassias stability of neutral stochastic functional differential equations

In this paper, by using the Gronwall inequality, we show two new results on the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of neutral stochastic functional differential equations. Two examples illustrating our results are exhibited.

A note on controllability of impulsive neutral stochastic functional differential equations driven by a Rosenblatt process in a Hilbert space

In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process in a Hilbert space. The controllability results are obtained by the Banach fixed point theorem. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients

<p style='text-indent:20px;'>In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.</p>

Existence and Stability of Solutions to Neutral Conformable Stochastic Functional Differential Equations

Existence and Stability of Solutions to Neutral Conformable Stochastic Functional Differential Equations

Stability analysis of theθ-method for hybrid neutral stochastic functional differential equations with jumps

• Few results on the stability of numerical methods for the neutral stochastic functional differential equations with Markovian switching and jumps, results obtained in this manuscript can enrich this area. • Illustrate the almost sure exponential stability and the mean-square exponential stability of the trivial solution. • Show the θ -method can preserve the almost sure exponential stability and the mean-square exponential stability of the trivial solution under some appropriate condition. • The almost sure exponential stability of the θ -method is obtained by the discrete semi-martingale convergence theorem without resorting to using the Borel-Cantelli lemma and the Chebyshev inequality. Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the θ -method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the θ -method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results.

Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

On exponential stability in mean square of neutral stochastic functional differential equations

General neutral stochastic functional differential equations are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of neutral stochastic functional differential equations are given. A discussion and illustrative examples are presented.

Controllability of fractional neutral functional differential equations with infinite delay driven by fractional Brownian motion

Abstract In this work, we establish a controllability result for a class of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion. To attain our objective we adapt the argument of Lakhel & McKibben (2018, Stochastics 90, no. 3, 313–329) where the existence of mild solutions to such stochastic equations was studied. An example is provided to show the applicability of the theoretical result.

Partial Practical Exponential Stability of Neutral Stochastic Functional Differential Equations with Markovian Switching

In this paper, we investigate the partial practical exponential stability of neutral stochastic functional differential equations with Markovian switching. The main tool used to prove the results is the Lyapunov method. We analyze an illustrative example to show the applicability and interest of the main results.

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

Averaging of neutral stochastic partial functional differential equations involving delayed impulses

In this paper, we initiate a study on averaging principles for neutral stochastic partial functional differential equations (NSPFDEs) with delayed impulses. With the aid of inequality techniques, the semigroup approach and some technical transformations, we derive an averaging principle for a class of NSPFDEs involving delayed impulses. The results obtained ensure that one can focus on the averaged autonomous system without impulses in place of the original system, thus providing an effective way for reducing complexity. In the end, an example is worked out to demonstrate the theoretical results.

β -Stability in q -th Moment of Neutral Impulsive Stochastic Functional Differential Equations with Markovian Switching

In this paper, we investigate the β -stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, β -stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.

Perturbed Impulsive Neutral Stochastic Functional Differential Equations

This paper studies the asymptotic behavior of the mild solution for a class of perturbed impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in Hilbert space. We establish the conditions under which the mild solutions of perturbed impulsive neutral stochastic functional differential equation and the unperturbed one are close on finite time interval when the perturbation tends to zero. Moreover, we show the result holds on time interval whose length tends to infinity as the perturbation tends to zero. As an application, the asymptotic behavior of the mild solution for a class of perturbed impulsive neutral stochastic partial differential equations driven by fractional Brownian motion in Hilbert space is proposed to show the feasibility of the obtained result.

Existence results for impulsive delayed neutral stochastic functional differential equations with noncompact semigroup

In this paper, we study the existence of mild solutions for impulsive neutral stochastic functional differential equations driven by a fractional Brownian motion with noncompact semigroup and varying-time delays in a Hilbert space. With the help of the Hausdorff measure of noncompactness, the Mönch fixed-point theorem and some inequality technique, some new criteria to guarantee the mild solution for impulsive neutral stochastic functional differential equations are obtained. Moreover, some previous results will be significantly improved in our paper.

EXISTENCE AND EXPONENTIAL STABILITY OF MILD SOLUTIONS FOR SECOND-ORDER NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATION WITH RANDOM IMPULSES

In this paper, we consider the existence and exponential stability in mean square of mild solutions to second-order neutral stochastic functional differential equations with random impulses in Hilbert space. Firstly, the existence of mild solutions to the equations is proved by using the noncompact measurement strategy and the Mönch fixed point theorem. Then, the mean square exponential stability for the mild solution of the considered equations is obtained by establishing an integral inequality. Finally, an example is given to illustrate our results.

Well-Posedness and Exponential Estimates for the Solutions to Neutral Stochastic Functional Differential Equations with Infinite Delay

AbstractIn this work, neutral stochastic functional differential equations with infinite delay (NSFD-EwID) have been addressed. By using the Euler-Maruyama scheme and a localization argument, the existence and uniqueness of solutions to NSFDEwID at the state space Cr under the local weak monotone condition, the weak coercivity condition and the global condition on the neutral term have been investigated. In addition, the L2 and exponential estimates of NSFDEwID have been studied.

Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulses

This paper aims at addressing the issue of a periodic averaging method for neutral stochastic functional differential equations with delayed impulses. Two periodic averaging theorems are presented and the approximate equivalence between the solutions to the original systems and those to the reduced averaged systems without impulses is proved. Further, we show a brief framework of extending our main results to Lévy case. At last, an example is given to demonstrate the procedure and validity of the proposed periodic averaging method.

The stability with general decay rate of neutral stochastic functional hybrid differential equations with Lévy noise

This paper is concerned with the existence and uniqueness, the almost sure stability with general decay rate (including almost sure exponential stability, almost sure polynomial stability and almost sure logarithmic stability) for the global solution of nonlinear neutral stochastic functional hybrid differential equations with Lévy noise. The key technique used is the method of Lyapunov function, nonnegative semi-martingale convergence theorem and the theory of M-matrix. We use auxiliary functions to dominate the corresponding Lyapunov function and the diffusion operator. Our conditions on the diffusion operator are weaker than those in the related existing works. Finally, one example is given to show the effectiveness of the obtained theory.