Nonlinear Stochastic Differential Delay Research Articles

Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Lévy jumps under different structures

This paper investigates the boundedness and stability of a class of nonlinear hybrid neutral stochastic differential delay systems with Lévy jumps and different structures. The coefficients in this system satisfy the local Lipschitz condition and a suitable Khasminskii-type condition, and the state space of the system is separated into two subsets, the existence uniqueness, asymptotic boundedness, and exponential stability of the system are obtained by designing a new Lyapunov function and applying the M-matrix technique as well as dealing with the non-differentiable delay function. Different with the existing work, we not only consider the neutral term, but also the case of the delay function being bounded and non-differentiable. At last, numerical examples are performed to manifest the obtained results.

Stabilization of Highly Nonlinear Hybrid Stochastic Differential Delay Equations with Lévy Noise by Delay Feedback Control

Stabilization of Highly Nonlinear Hybrid Stochastic Differential Delay Equations with Lévy Noise by Delay Feedback Control

Stability analysis of highly nonlinear hybrid stochastic systems with Poisson jump

This article discusses a class of nonlinear hybrid stochastic differential delay equations with Poisson jump and different structures. Compared with the Brownian motion, the jump makes the analysis more complex by reason of the discontinuity of its sample paths. Moreover, the coefficients meet a novel nonlinear growth condition and different structures in different switch modes. By using M-matrices and Lyapunov functions, we prove that the existence-uniqueness, asymptotic boundedness and exponential stability of the solution. Finally, we give two examples to demonstrate the usefulness of our theory.

Advances in nonlinear hybrid stochastic differential delay equations: Existence, boundedness and stability

This paper is concerned with a class of highly nonlinear hybrid stochastic differential delay equations (SDDEs). Different from the most existing papers, the time delay functions in the SDDEs are no longer required to be differentiable, not to mention their derivatives are less than 1. The generalized Hasminskii-type theorems are established for the existence and uniqueness of the global solutions. Comparing with the existing results, we show our new theorems are much more general and can be applied to a much wider class of highly nonlinear SDDEs. Further sufficient conditions are also obtained for the asymptotic boundedness and stability.

Boundedness and stability of nonlinear hybrid stochastic differential delay equation disturbed by Lévy noise

This study focuses on the boundedness and stability of the nonlinear hybrid stochastic differential delay equation (NHSDDE) disturbed by Lévy noise (LN). Different from the previous systems with Markovian switching, not only different parameters under different modes, but also different structures under different modes are considered, where a novel Lyapunov function is constructed to handle the impacts of different structures under different modes. Without the linear growth condition, the existence of the global solution is proved for the NHSDDE with LN. Then, the M-matrix technique is applied to build some reasonable theorems upon the vth moment asymptotic boundedness, the vth moment exponential stability and the almost surely exponential stability for the NHSDDE driven by LN. An approach to solving the admissible upper bound of the time-delay derivative is given for guaranteeing the asymptotic boundedness and the exponential stability of the global solution. Finally, two numerical examples and homologous simulated outcomes are provided to indicate the usefulness of the presented theoretical findings.

The truncated θ-Milstein method for nonautonomous and highly nonlinear stochastic differential delay equations

In this paper we propose and analyze a truncated θ-Milstein method for solving a class of non-autonomous stochastic differential delay equations with highly nonlinear coefficients. Under a generalized monotone condition of the coefficients, the convergence order of the truncated θ-Milstein method is obtained. We find that the convergence rate is dependent on the exponent of the Hölder conditions. To verify our theoretical findings, we provide a numerical example.

Advances in stabilization of highly nonlinear hybrid delay systems

Given an unstable highly nonlinear hybrid stochastic differential delay equation (SDDE, also known as an SDDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDDE become exponentially stable? Recent work by Li and Mao in 2020 gave a positive answer when the delay in the given SDDE is a positive constant. It is also noted that in their paper the time lag in the feedback control is another constant. However, time delay in a real-world system is often a variable of time while it is difficult to implement the feedback control in practice if the time lag involved is a strict constant. Mathematically speaking, the stabilization problem becomes much harder if these delays are time-varying, in particular, if they are not differentiable. The aim of this paper is to tackle the stabilization problem under non-differentiable time delays. One more new feature in this paper is that the feedback control function used is bounded.

Stabilization by variable-delay feedback control for highly nonlinear hybrid stochastic differential delay equations

We consider a hybrid SDE (stochastic differential equation) with variable delays, which is unstable and its coefficients satisfy some polynomial growth conditions. We want to find a variable-delays feedback control such that the controlled hybrid SDE with variable delay becomes stable. We establish some novel stability criteria of the controlled SDE with variable delays. Our results improve the existing results from hybrid SDEs with constant-delays to hybrid SDEs with variable-delays. Finally, an example is given to confirm our results.

The Truncated Theta-EM Method for Nonlinear and Nonautonomous Hybrid Stochastic Differential Delay Equations with Poisson Jumps

In this paper, we study a class of nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs). The convergence rate of the truncated theta-EM numerical solutions to HSDDEwPJs is investigated under given conditions. An example is shown to support our theory.

Numerical method of highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching

In this paper, we establish a partially truncated Euler–Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.

Almost sure exponential stability of semi-Euler numerical scheme for nonlinear stochastic functional differential equation

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out.

Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Guo et al. [GMY17] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations(SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [GMY17] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control

Given an unstable hybrid stochastic differential equation (SDDE, also known as an SDDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDDE become asymptotically stable? If the feedback control is based on the current state, the stabilisation problem has been studied. However, there is little known when the feedback control is based on the past state. The problem becomes even harder when the coefficients of the underlying hybrid SDDE do not satisfy the linear growth condition (namely, the coefficients are highly nonlinear). The aim of this paper is to tackle the stabilisation problem for a given unstable highly nonlinear hybrid SDDE.

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.

On the asymptotic stability of solutions of stochastic differential delay equations of second order

ABSTRACTIn this paper, we consider a non-linear stochastic differential delay equation (SDDE) of second order. We derive new sufficient conditions which guarantee stochastically stability and stochastically asymptotically stability of the zero solution of that SDDE. Here, the technique of the proof is based on the definition of a suitable Lyapunov-Krasovskii functional, which gives meaningful results for the problem under consideration. The derived results extend and improve some result of in the relevant literature, which are related to the qualitative properties of solutions of a SDDE of second order. The results of this paper are new and have novelty, and they do a contribution to the topic and relevant literature. As an application, an example is given to show the effectiveness and applicability of the obtained results. Finally, by the results of this paper, we extend and improve some recent results that can be found in the relevant literature.

Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

Convergence rate of the truncated Milstein method of stochastic differential delay equations

This paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations (SDDEs) without the linear growth condition. On the one hand, these nonlinear SDDEs do not have explicit solutions, therefore implementable numerical methods for such SDDEs are required. On the other hand, the implicit Euler methods are known to converge strongly to the exact solution of such SDDEs. However, they require additional computational efforts. In this article, we propose the truncated Milstein method which is an explicit method under the local Lipschitz condition plus Khasminskii-type condition, study its pth monent boundedness (p is a parameter in Khasminskii-type condition) and show that its rate of strong convergence is close to one.

Stabilization and destabilization of nonlinear stochastic differential delay equations

ABSTRACTThis present paper deals with the problem of stabilization and destabilization of nonlinear stochastic differential delay equations. The key techniques used in this paper are the LaSalle-type stability theorems, the nonnegative semimartingale convergence theorem, and the law of large numbers for martingales. The proposed results can be applied to study the stabilization and destabilization of more general nonlinear stochastic dynamical time-delay systems. Computer simulations are presented to illustrate our theory.

Generalized criteria on delay‐dependent stability of highly nonlinear hybrid stochastic systems

SummaryOur recent paper (Fei W, etal. Delay dependent stability of highly nonlinear hybrid stochastic systems. Automatica. 2017;82:165‐170) is the first to establish delay‐dependent criteria for highly nonlinear hybrid stochastic differential delay equations (SDDEs) (by highly nonlinear, we mean that the coefficients of the SDDEs do not have to satisfy the linear growth condition). This is an important breakthrough in the stability study as all existing delay stability criteria before could only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In this continuation, we will point out one restrictive condition imposed in our earlier paper. We will then develop our ideas and methods there to remove this restrictive condition so that our improved results cover a much wider class of hybrid SDDEs.

The pth moment exponential stability and almost surely exponential stability of stochastic differential delay equations with Poisson jump

This paper is devoted to study a class of nonlinear stochastic differential delay equations with Poisson jump. In comparison to the Brownian motion, the jump leads to the discontinuity of sample paths, which makes the analysis more difficult. We first introduce the local Lipschitz condition and a new nonlinear growth condition, which is weaker than those in the previous literature. Then by virtue of Lyapunov function and semi-martingale convergence theorem, we prove that the considered stochastic system has a unique global solution. Moreover, we also investigate the pth moment exponential stability and the almost surely exponential stability of solutions. Finally, an example is given to illustrate the effectiveness of our theoretical results.