Neutral Stochastic Differential Equations Research Articles

Exponential stability of neutral stochastic delay differential equation with delay-dependent impulses

In this paper, we mainly consider exponential stability (ES) for a type of neutral stochastic delay differential equations with delay-dependent impulses (NSDDEWDI). By means of the average dwell time (ADT) condition, a new sufficient criterion for ES is given. The infinitesimal operator of Lyapunov functions is controlled by two time-variant coefficients. Furthermore, these new conditions are calculated through an example so as to verify the validity of our results.

Stabilisation by delay feedback control for highly nonlinear neutral stochastic differential equations

Given an unstable hybrid neutral stochastic differential equation (NSDE), can we design a delay feedback control to make the controlled hybrid NSDE become stable? It has been proved that this is possible under the linear growth condition. However, there is no answer to the question if the drift and diffusion coefficients of the given unstable NSDE satisfy highly nonlinear growth condition. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid NSDEs whose coefficients satisfy the polynomial growth condition.

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.

Neutral fractional stochastic partial differential equations with Clarke subdifferential

ABSTRACT By using fractional calculus, stochastic analysis theory and fixed point theorems, sufficient conditions for approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential are established. Finally, an example is given to illustrate the obtained results.

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.

Global attracting set and exponential decay of coupled neutral SPDEs driven by fractional Brownian motion

In this paper, we are concerned with a class of coupled neutral stochastic partial differential equations driven by fractional Brownian motion with Hurst parameter . On account of the Perov's fixed point theorem and semigroup theory, we prove the existence and uniqueness of mild solution. Subsequently, by using delay integral inequalities, we identify the global attracting sets of the equations under investigation. Furthermore, we obtain some sufficient conditions that ensure the exponential decay of mild solutions in the pth moment. Lastly, we present an example to illustrate our theory in this work.

<i>G</i>-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by \begin{document}$ G $\end{document} -Brownian motion (VNSDDEGs in short) under global Caratheodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in \begin{document}$ L^p $\end{document} -sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.

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.

Convergence of the EM method for NSDEs with time-dependent delay in the G-framework

ABSTRACTConsider a neutral stochastic differential equation (NSDE) with time-dependent delay () in the G-framework where denotes a G-Brownian motion and the quadratic variation process of . We introduce an Euler–Maruyama (EM) method for solving this equation and prove that the EM approximate solution converges to the exact solution with a strong order of the mean square closeness equal to one under the global Lipschitz condition. A numerical example is provided to illustrate the effectiveness of our 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.

Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main 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.

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

SummaryIn this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.

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.

The existence and Hyers\u2013Ulam stability of solution for an impulsive Riemann\u2013Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1

This paper deals with the existence of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1<beta<2 and its Hyers–Ulam stability. We prove the mild solutions for the equation using basic theorems of fractional differential equation. The existence result of the equation is obtained by Mönch’s fixed point theorem. Finally, we prove the Hyers–Ulam stability of the solution.

Almost sure exponential stability of the $$\theta $$ θ -Euler–Maruyama method, when $$\theta \in (\frac{1}{2},1)$$ θ ∈ ( 1 2 , 1 ) , for neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions

This paper is motivated by Lan and Yuan (J Comput Appl Math 285:230–242, 2015). The main aim of this paper is to establish certain results for the $$\theta $$ -Euler–Maruyama method for a class of neutral stochastic differential equations with time-dependent delay. The method is defined such that, in general case, it is implicit in both drift and neutral term. The drift and neutral term are both parameterized by $$\theta $$ in a way which guarantees that for $$\theta =0$$ and $$\theta =1$$ , the method reduces to the Euler–Maruyama method and backward Euler method, respectively, which can be found in the literature. The one-sided Lipschitz conditions in the present-state and delayed arguments of the drift coefficient of this class of equations for any $$\theta \in [0,1]$$ are employed in order to guarantee the existence and uniqueness of the appropriate $$\theta $$ -Euler–Maruyama approximate solution. The main result of this paper is almost sure exponential stability of the $$\theta $$ -Euler–Maruyama method, for $$\theta \in (\frac{1}{2},1),$$ under nonlinear growth conditions. Some comments and conclusions are presented for the corresponding deterministic case. An example and numerical simulations are provided to support the main results of the paper.

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.

Exponential Stability of Highly Nonlinear Neutral Pantograph Stochastic Differential Equations

AbstractIn this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M‐matrix are used to study exponential stability and boundedness of the hybrid NPSDEs.