Linear Stochastic Delay Research Articles

Analysis of stability for stochastic delay integro-differential equations

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler–Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

A note on exponential stability of non-autonomous linear stochastic differential delay equations driven by a fractional Brownian motion with Hurst index [formula omitted

We prove a criterion for the almost sure exponential stability of the scalar non-autonomous linear stochastic differential delay equations driven by a fractional Brownian motion with Hurst index >1∕2. To do that, we need a version of existence and uniqueness of the solution for non-autonomous linear system.

Stability of linear stochastic delay differential equations with infinite Markovian switchings

SummaryThis paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.

One-parameter statistical model for linear stochastic differential equation with time delay

ABSTRACTAssume that we observe a stochastic process , which satisfies the linear stochastic delay differential equation where a is a finite signed measure on . The local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of , local asymptotic quadraticity is shown if , and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if , where is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator of based on can be derived as.

Exponential Mean Square Stability of the BE Method for Linear Neutral Hybrid Stochastic Delay Differential Equations

Exponential Mean Square Stability of the BE Method for Linear Neutral Hybrid Stochastic Delay Differential Equations

Sliding mode mean square filter design for linear stochastic time delay systems

This paper investigates the mean-square filtering problem for a linear time delay systems with Gaussian white noises. The obtained solution contains a sliding mode term, signum of the innovations process. It is demonstrated that the estimate produced by the designed filter generates the mean-square estimate, which has the same minimum estimation-error variance as the best estimate given by the classical Kalman–Bucy filter. The theoretical result is applied to an illustrative example: the tryptophan operon of E. coli, verifying the performance of the designed filter. It is demonstrated that the estimates produced by the designed sliding-mode mean-square filter and the Kalman–Bucy filter yield the same estimation-error variance. Simulation graphs demonstrate the better performance of the designed sliding-mode filter and show the potential of the proposed new filter.

Numerical stability analysis of linear stochastic delay differential equations using Chebyshev spectral continuous time approximation

In this paper, a numerical approach is developed for the stability analysis of linear stochastic delay differential equations (SDDEs) in the parameter space based on the Chebyshev Spectral Continuous Time Approximation (CSCTA) technique. The CSCTA method is used to approximate an infinite-dimensional linear SDDE as a set of linear stochastic differential equations (SDEs). The mean and mean-square stability concepts are employed for the stochastic stability analysis of the resulting SDE. For this purpose, a set of linear deterministic differential equations for both the first and second moments are obtained using the Ito differential rule. Two examples are provided: a first order SDDE with multiplicative stochastic excitation and a second order SDDE with both additive and multiplicative stochastic excitation. In both examples the stability charts obtained from the proposed approach match those obtained using the stochastic semi-discretization method as described by Elbeyli et al. (Commun Nonlinear Sci Numer Simul 10(1):85–94, 2005). In the first example the stability results obtained from both numerical approaches are found to be less conservative than the Lyapunov-based stability region obtained by Samiei et al. (Int J Dyn Control 1(1):64–80, 2013).

Convergence and stability of Euler method for impulsive stochastic delay differential equations

This paper deals with the mean square convergence and mean square exponential stability of an Euler scheme for a linear impulsive stochastic delay differential equation (ISDDE). First, a method is presented to take the grid points of the numerical scheme. Based on this method, a fixed stepsize numerical scheme is provided. Based on the method of fixed stepsize grid points, an Euler method is given. The convergence of the Euler method is considered and it is shown the Euler scheme is of mean square convergence with order 1/2. Then the mean square exponential stability is studied. Using Lyapunov-like techniques, the sufficient conditions to guarantee the mean square exponential stability are obtained. The result shows that the mean square exponential stability may be reproduced by the Euler scheme for linear ISDDEs, under the restriction on the stepsize. At last, examples are given to illustrate our results.

Second moment boundedness of linear stochastic delay differential equations

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework--for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay--of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a specific case of a type of $2$-dimensional equation that the stochastic terms are decoupled. For the $2$-dimensional equation, we obtain the characteristic function that is explicitly given by equation coefficients, and the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.

Moment boundedness of linear stochastic delay differential equations with distributed delay

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.

An analysis of the exponential stability of linear stochastic neutral delay systems

SummaryThis paper is concerned with the analysis of the mean square exponential stability and the almost sure exponential stability of linear stochastic neutral delay systems. A general stability result on the mean square and almost sure exponential stability of such systems is established. Based on this stability result, the delay partitioning technique is adopted to obtain a delay‐dependent stability condition in terms of linear matrix inequalities (LMIs). In obtaining these LMIs, some basic rules of the Ito calculus are also utilized to introduce slack matrices so as to further reduce conservatism. Some numerical examples borrowed from the literature are used to show that, as the number of the partitioning intervals increases, the allowable delay determined by the proposed LMI condition approaches hmax, the maximal allowable delay for the stability of the considered system, indicating the effectiveness of the proposed stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.

A novel result on stability analysis for uncertain neutral stochastic time-varying delay systems

This paper is concerned with the mean-square exponential stability analysis for uncertain neutral linear stochastic time-varying delay systems. By Lyapunov–Krasovskii theory and linear matrix inequality method, under the generalized Finsler lemma (GFL) framework, delay-dependent mean-square exponential stability criteria are established without involving model transformation, cross-terms bounding technique or additional free-weighting matrix. Moreover, GFL is also employed to obtain stability criteria for a class of uncertain linear stochastic neutral systems with different discrete and neutral delays. Numerical examples are presented to verify that the proposed approach is both less conservative and less computationally complex than the existing results.

DYNAMICS OF DETERMINISTIC AND STOCHASTIC EVOLUTIONARY GAMES WITH MULTIPLE DELAYS

In the present paper, we study the effect of time delays in evolutionary games with one population of users and two strategies. The case where the delays, corresponding to different strategies, are not the same is considered. The local stability of the stationary state for the replicator dynamics is analyzed. We show that there is Hopf bifurcation. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and normal form theory. The stochastic evolutionary game with delay is taken into consideration. We also study the behavior of the first and second solution moments for linear stochastic differential delay equation in the presence of white and colored noise. The last part of the paper includes numerical simulations and conclusions.

Dynamic Games for Stochastic Systems with Delay

AbstractThis paper discusses dynamic games for a class of linear stochastic delay systems governed by Itô's stochastic differential equation. The Pareto and Nash strategies are developed by solving cross‐coupled matrix inequalities. To obtain these strategy sets, new cross‐coupled algebraic equations (CSAEs) are established on the basis of the Karush‐Kuhn‐Tucker (KKT) conditions, which constitute the necessary conditions. It is noteworthy that the state feedback strategies can be obtained by solving the linear matrix inequality (LMI) recursively. Finally, a numerical example showing the effectiveness of the proposed methods and the attained cost bounds is described.

On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations

We are concerned with the exponential mean-square stability of two-step Maruyama methods for stochastic differential equations with time delay. We propose a family of schemes and prove that it can maintain the exponential mean-square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation. Numerical results for linear and nonlinear equations show that this family of two-step Maruyama methods exhibits better stability than previous two-step Maruyama methods.

Exponential growth rate for a singular linear stochastic delay differential equation

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space)of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as longas it is not identically zero. Due to the singular nature of the equation this property does not follow fromavailable results on stochastic delay differential equations. The key technique is to establish existence and uniquenessof an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).

A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations

The stability of equilibrium solutions of a deterministic linear system of delay differential equationscan be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based onthe vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providingsufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system andthen compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.

Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.

A note on stability of the split-step backward Euler method for linear stochastic delay integro-differential equations

In the literature (Tan and Wang, 2010), Tan and Wang investigated the convergence of the split-step backward Euler (SSBE) method for linear stochastic delay integro-differential equations (SDIDEs) and proved the mean-square stability of SSBE method under some condition. Unfortunately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.

T-stability of numerical solutions for linear stochastic differential equations with delay

In this paper, T-stability of the Euler-Maruyama method is taken into account for linear stochastic delay differential equations with multiplicative noise and constant time lag in the Itosense. Under a certain condition for coefficients, T-stability of the numerical scheme is researched. Moreover, some numerical examples will be presented to support the theoretical results.