Neutral Stochastic Differential Delay Equations Research Articles

Stability analysis of neutral stochastic differential delay equations driven by Lévy noises

This paper mainly analyzes the well-posedness, and the stability analysis for the global solution of neutral stochastic differential delay equations (NSDDEs) driven by Lévy noises. By using an integral lemma and a Lyapunov function approach, the existence and uniqueness theorem is proved. Then, by using the inequality technique and the stochastic analysis theory, the exponential stability in pth(p ≥ 2) moment of such equations is discussed. By using another integral lemma, and using the Baralat lemma as well as the stochastic analysis, the almost surely asymptotic stability is also studied. Finally, one example is given to check the effectiveness of the findings derived.

Convergence rates of theta-method for NSDDEs under non-globally Lipschitz continuous coefficients

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of [Formula: see text]-EM schemes are given for these equations driven by Brownian motion and pure jumps, respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.

Numerical analysis of three θ methods for neutral stochastic delay differential equations

This article focus on the numerical analysis of stochastic θ-method, split-step θ method and one-leg θ method for neutral stochastic differential equations with time lag. When some stability conditions are met, the three θ methods turned out to be mean-square stable asymptomatically. From the numerical analysis, we can see that the asymptomatically mean-square stability for both linear and nonlinear stochastic differential equations depends on the step size h and parameter θ, which numerical analysis are based on stochastic θ-method, split-step θ method and one-leg θ method.

Boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays

This paper reports the boundedness and stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs) with multiple delays. Without imposing linear growth condition, the boundedness and exponential stability of the exact solution are investigated by Lyapunov functional method. In particular, using the M-matrix technique, the mean square exponential stability is obtained. Finally, three examples are presented to verify our results.

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

The aim of this paper is to investigate strong convergence of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations introduced in Lan (2018). Strong convergence rates of the given numerical scheme to the exact solutions at fixed time T are obtained under local Lipschitz and Khasminskii-type conditions. Moreover, convergence rates over a time interval [0,T] are also obtained under additional polynomial growth condition on g without the weak monotonicity condition (which is usually the standard assumption to obtain the convergence rate). Two examples are presented to interpret our conclusions.

On the Asymptotic Behavior for Neutral Stochastic Differential Delay Equations

This note investigates the existence and uniqueness as well as the stability of the general decay rate of the global solution for neutral stochastic differential equations with time-varying delay under a locally Lipschitz condition, a contractive condition, and a monotonicity condition. The stability results are derived by using the Lyapunov function approach and some stochastic analysis techniques, which not only cover the exponential stability in the $p$ th $(p>0)$ -moment and the almost sure exponential stability, but also the polynomial stability in the $p$ th $(p>0)$ -moment and the almost sure polynomial stability. Two examples including one coupled system consisting of a mass–spring–damper connected to a pendulum and the nonlinear external random force are given to illustrate the effectiveness of the obtained results.

Switching-dominated stability of numerical solutions for hybrid neutral stochastic differential delay equations

This paper is concerned with the exponential stability of numerical solutions for neutral stochastic differential delay equations (NSDDEs) with Markovian switching (or hybrid NSDDEs). It is known that Markov chain can work as a stabilizing factor, that is, when some subsystems are stable and others are unstable, the overall system can be stable. In this paper, such property is called switching-dominated stability, and we prove that the switching-dominated stability can be recovered in the stability of numerical solutions based on the switched Lyapunov function method. Firstly, a fundamental stability criterion is established for the numerical solutions. Then under a linear growth condition, we show that the Euler–Maruyama (EM) method can share the mean square exponential stability of the exact solution. When the linear growth condition is defied, but a one-sided Lipschitz condition is satisfied, we show that the backward EM (BEM) method can reproduce the mean square exponential stability of the exact solution. Numerical experiments are carried out to confirm the theoretical results.

A New Stability Criterion for Neutral Stochastic Delay Differential Equations with Markovian Switching

In this short paper, a new stability theorem for neutral stochastic delay differential equations with Markovian switching is investigated by applying stochastic analysis technique and Razumikhin stability approach. A novel criterion of thepth moment exponential stability is derived for the related systems. The feature of the criterion shows that the estimated upper bound for the diffusion operator of Lyapunov function is allowed to be indefinite, even if to be unbounded, which can loosen the constraints of the existing results. Last, an example is provided to illustrate the usefulness and significance of the theoretical results.

Stability results for neutral stochastic functional differential equations via fixed point methods

ABSTRACT In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo, Y., Xu, C., & Wu, J. [(2017). Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle. International Journal of Control, 90, 1555–1560]. Finally, an example is exhibited to illustrate the effectiveness of the proposed results.

Strong convergence of a tamed theta scheme for NSDDEs with one-sided Lipschitz drift

This paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one-sided Lipschitz drift. Strong convergence rate is revealed under a global one-sided Lipschitz condition, while for a local one-sided Lipschitz condition, the tamed theta scheme is modified to ensure the well-posedness of implicit numerical schemes, then we show the convergence of the numerical solutions.

Almost sure convergence rate of θ-EM scheme for neutral SDDEs

This paper is concerned with almost sure convergence of θ -EM scheme for neutral stochastic differential delay equations under local Lipschitz conditions with θ ∈ 1 2 , 1 . It is revealed that the strong convergence rate under local Lipschitz condition is 1/2, while almost sure convergence rate is less than 1/2, provided the local Lipschitz constant, valid on balls of radius R , grows not faster than log R .

Stability of highly nonlinear neutral stochastic differential delay equations

Stability criteria for neutral stochastic differential delay equations (NSDDEs) have been studied intensively for the past several decades. Most of these criteria can only be applied to NSDDEs where their coefficients are either linear or nonlinear but bounded by linear functions. This paper is concerned with the stability of hybrid NSDDEs without the linear growth condition, to which we will refer as highly nonlinear ones. The stability criteria established in this paper will be dependent on delays.

Asymptotic exponential stability of modified truncated EM method for neutral stochastic differential delay equations

The aim of this paper is to investigate the asymptotic mean–square and almost sure exponential stability of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations under non linear conditions. It is proved that the so called modified truncated Euler–Maruyama method is mean–square (almost surely) exponentially stable under suitable (local Lipschitz and Khasminskii-type) conditions. An example is provided to support our conclusions.

Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by G-Brownian motion

There are few numerical analysis results for nonlinear neutral stochastic delay differential equations driven by G-Brownian motion (G-NSDDEs). This paper is concerned with the numerical solutions of the G-NSDDEs to fill this gap. In this paper, we first devote to show that the stochastic theta numerical solution converges to the exact solution for the G-NSDDE. We then prove that the backward Euler–Maruyama numerical solution for the G-NSDDE is asymptotically mean-square stable under suitable conditions. Moreover, a numerical example is demonstrated to illustrate the effectiveness of our obtained results.

Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler–Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Tamed EM scheme of neutral stochastic differential delay equations

In this paper, we investigate the convergence of the tamed Euler-Maruyama (EM) scheme for a class of neutral stochastic differential delay equations. The strong convergence results of the tamed EM scheme are presented under global and local non-Lipschitz conditions, respectively. Moreover, under a global Lipschitz condition, we provide the rate of the convergence of tamed EM, which is smaller than the rate convergence of classical EM scheme one half.

Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations with Poisson Jumps

Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations with Poisson Jumps

Strong convergence of the split-step theta method for neutral stochastic delay differential equations

Neutral stochastic delay differential equations often appear in various fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step theta (SST) method for the neutral stochastic delay differential equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the SST method with θ∈[0,1] strongly converges to the exact solution with the order 12. Some numerical results are presented to confirm the obtained results.

Moderate deviations for neutral stochastic differential delay equations with jumps

A moderate deviation principle for neutral stochastic differential delay equations driven by Poisson random measure is established. The weak convergence method introduced by Budhiraja et al. (2016) plays a key role.

Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle

ABSTRACTIn this paper, we consider a class of neutral stochastic differential systems with variable delays and give sufficient conditions to ensure that the zero solution is asymptotically stable in the mean square by means of a generalisation of Banach's contraction principle. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. The results are shown to improve the previous globally stable results derived in the literature.