Stochastic Differential Delay Research Articles

Stability of stochastic dynamic equations with time-varying delay on time scales

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for -delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.

Exponential ultimate boundedness of impulsive stochastic delay difference systems

SummaryThis paper is concerned with the exponential ultimate boundedness problems for the impulsive stochastic delay difference systems. Several sufficient conditions on the global pth moment exponential ultimate boundedness are presented by using the Lyapunov methods and the algebraic inequality techniques, and the estimated exponential convergence rate and the ultimate bound are provided as well. As an application, the boundedness criteria are applied to a class of discrete impulsive stochastic neural networks with delays. The obtained results show that the impulses not only can stabilize an unstable stochastic difference delay system but also can make an unbounded stochastic difference delay system into a bounded system. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.

RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

A posteriori analysis of finite element discretizations of stochastic partial differential delay equations

In this paper, we study a posteriori error estimates for finite element approximation of stochastic partial differential delay equations containing a noise. We derive an energy norm a posteriori bounds for an Euler time-stepping method combined with a standard Galerkin schemes for the problems. For accessibility, we first address the spatially semidiscrete case and then move to the fully discrete scheme.

Stability and Stabilization of Impulsive Stochastic Delay Difference Equations

When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov‐Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

General Decay Stability for Stochastic Functional Differential Equations with Infinite Delay

So far there are not many results on the stability for stochastic functional differential equations with infinite delay. The main aim of this paper is to establish some new criteria on the stability with general decay rate for stochastic functional differential equations with infinite delay. To illustrate the applications of our theories clearly, this paper also examines a scalar infinite delay stochastic functional differential equations with polynomial coefficients.

Stochastic Delay Difference and Differential Equations: Applications to Financial Markets

This thesis deals with the asymptotic behaviour of stochastic difference and functional differential equations of Itˆo type. Numerical methods which both minimise error and preserve asymptotic features of the underlying continuous equation are studied. The equations have a form which makes them suitable to model financial markets in which agents use past prices. The second chapter deals with the behaviour of moving average models of price formation. We show that the asset returns are positively and exponentially correlated, while the presence of feedback traders causes either excess volatility or a market bubble or crash. These results are robust to the presence of nonlinearities in the traders’ demand functions. In Chapters 3 and 4, we show that these phenomena persist if trading takes place continuously by modelling the returns using linear and nonlinear stochastic functional differential equations (SFDEs). In the fifth chapter, we assume that some traders base their demand on the difference between current returns and the maximum return over several trading periods, leading to an analysis of stochastic difference equations with maximum functionals. Once again it is shown that prices either fluctuate or undergo a bubble or crash. In common with the earlier chapters, the size of the largest fluctuations and the growth rate of the bubble or crash is determined. The last three chapters are devoted to the discretisation of the SFDE presented in Chapter 4. Chapter 6 highlights problems that standard numerical methods face in reproducing long–run features of the dynamics of the general continuous–time model, while showing these standard methods work in some cases. Chapter 7 develops an alternative method for discretising the solution of the continuous time equation, and shows that it preserves the desired long–run behaviour. Chapter 8 demonstrates that this alternative method converges to the solution of the continuous equation, given sufficient computational effort.

Effect of small time delay in a predator-prey model within random environment

We consider a non-Kolmogorov type predator-prey model with and without delay. The local stability and Hopf bifurcation results are stated for both the cases of the deterministic system. The effect of stochastic perturbation in the form of parametric white and colored noise are considered. The exponential mean square stability of the trivial solutions for the stochastic differential equations and stochastic delay differential equations are studied under the Ito interpretation. To study the exponential mean square stability of the interior equilibrium point for the delayed model system in the presence of parametric white and colored noise, an approximation procedure is followed for the system of stochastic delay differential equation model system under the assumption that the magnitude of time delay is small. We have obtained threshold magnitude of discrete time delay. Critical magnitude for noise intensities are obtained for stochastic stability aspects. Numerical simulation results are provided in support of our analytical findings.

On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term

Global asymptotic stability conditions for discrete vector nonlinear stochastic systems with state delay and Volterra diffusion term are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The derived stability conditions are directly expressed in terms of the system coefficients. A number of nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given. The obtained results are compared to some previously known asymptotic stability conditions for discrete nonlinear stochastic systems.

Exponential stability of a kind of stochastic delay difference equations

We present a Razumilchin‐type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic “trade cycle” with the environmental noise.

Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

One concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data; there are other stability concepts, notably those concerned with persistent perturbations. Results are presented on the stability in p-th mean of solutions of stochastic delay differential equations with multiplicative noise, and of stochastic delay difference equations. The difference equations are of a type found in numerical analysis and we employ our results to obtain mean-square stability criteria for the solution of the Euler– Maruyama discretization of stochastic delay differential equations. The analysis proceeds as follows: We show that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution. We then produce a discrete analogue of the Halanay-type theory, that permits us to develop a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.

Exponential stability of stochastic differential (delay) equations with Markovian switching

We study stochastic differential equations and stochastic delay differential equations with Markovian switching, some sufficient conditions for the any pth monment exponential stability are obtained. Relax the necessary condition that LV is negative defnition in the known references.

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).

On a stochastic delay difference equation with boundary conditions and its Markov property

In the present paper we consider the one-dimensional stochastic delay difference equation with boundary condition X n+1=X n+F(X n)+g(X n−1)+ξ n,X 0=ψ(X N), n ∈ {0,…, N − 1}, N ⩾ 8 (where g( X −1) ≡ 0). We prove that under monotonicity (or Lipschitz) conditions over the coefficients f, g and ψ, there exists a unique solution { Z 1,…, Z N } for this problem and we study its Markov property. The main result that we are able to prove is that the two-dimensional process {( Z n , Z n+1 , 1 ⩽ n ⩽ N − 1} is a reciprocal Markov chain if and only if both the functions f and g are affine.