Linear Stochastic Delay Research Articles

Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation

The paper deals with convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. It is proved that the semi-implicit Euler method is convergent with strong order p= 1 2 . The conditions under which the method is MS-stable and GMS-stable are determined and the numerical experiments are given.

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.

An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays

Based on the linear matrix inequality method, we introduce the robust stability of uncertain linear stochastic differential delay systems with delay dependence. The parameter uncertainty is norm-bounded and the delays are time varying. We then extend the proposed theory to discuss the robust stabilization of uncertain stochastic differential delay systems.

Noise Induced Oscillation in Solutions of Stochastic Delay Differential Equations

This paper studies the oscillatory properties of solutions of linear scalar stochastic delay differential equations with multiplicative noise. It is shown that such noise will induce an oscillation in the solution whenever there is negative feedback from the delay term. The zeros of the process are a countable set; the solution is differentiable at each zero, and the zeros are simple. The addition of such noise does not alter the positivity of solutions when there is positive feedback.

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).

Strong discrete time approximation of stochastic differential equations with time delay

The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic differential equations (SDEs) with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given.

Robust stability of uncertain stochastic differential delay equations

In this paper we first discuss the robust stability of uncertain linear stochastic differential delay equations. We then extend the theory to cope with the robust stability of uncertain semi-linear stochastic differential delay equations. We shall also give several examples to illustrate our theory.

Solution moment stability in stochastic differential delay equations.

We study the behavior of the first and second solution moments for linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. In the presence of additive noise (white or colored), the stability domain of both moments is identical to that of the unperturbed system. When these moments lose stability, there is a Hopf bifurcation and the first moment oscillates with a period identical to the solution of the unperturbed equation, while the oscillation period of the second moment is exactly one half the period of the unperturbed solution and the first moment. When perturbations are of the parametric (or multiplicative) type and white noise is assumed, under the It\^o interpretation the first moment of the solution preserves properties of the solution of the deterministic equation, while the behavior of the second moment depends on the amplitude of the stochastic perturbation. The critical delay value at which the second moment loses stability and becomes oscillating is derived, and it is less than the critical delay for the first moment. Under the Stratonovich interpretation, quite different properties were observed for the moment equations, namely, various critical values of the delay and period of oscillations. For the case of parametric colored noise perturbations, sufficient (p-stability) conditions are derived which are independent of the value of delay, and it is shown that colored noise has a stabilizing effect with respect to white noise.

Noise and stability in differential delay equations

We study the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. Using a stochastic analog of the second Liapunov method, sufficient conditions for mean square and stochastic stability are derived.

Stability of linear stochastic delay under steady perturbations

Stability of linear stochastic delay under steady perturbations

Hyperbolic State Space Decomposition for a Linear Stochastic Delay Equation

We consider the equation $dX(t) = (\int_{ - r}^0 {X(s + t)d\mu (s))dt} + GdW(t)$ where $r > 0$, $\mu $ is a matrix-valued signed measure on $[ - r,0]$, W is an n-dimensional Wiener process and G is a fixed $n \times n$ matrix. Using the known decomposition $C = S \oplus U$ of $C = C[ - r,0]$ into a stable closed subspace S and a finite dimensional unstable subspace U for the analogous deterministic equation, we prove that the projection of the solution X onto S converges exponentially fast to a stationary Gaussian process. Also the speed of explosion of the projection onto U is exhibited.

Nonlinear flows of stochastic linear delay equations

The sample non-linearity of the classical Wiener integral as a function of continuous integrands is pointed out. As an application it is shown that the solution of a linear stochastic delay equation is an almost surely non-linear function of the initial trajectory segment.

Constrained stochastic controllability of infinite-dimensional linear systems

The main purpose of this article is to investigate the problem of (ε, δ)-stochastic controllability for linear systems of evolution type in infinite-dimensional spaces, wherein the controls are subjected to norm-bounded constrained sets. Some basic prerequisites of infinite-dimensional measures, in particular, Gaussian distributed type, are discussed. Corresponding to this measure, various properties of (ε, δ)-stochastic attainable sets in Hilbert spaces are studied. Necessary and sufficient conditions for (ε, δ)-stochastic controllability with respect to Hilbert space valued linear systems are obtained. Relationships with the deterministic counterpoint are noted. Pursuit game problems are also considered. Examples on systems governed by stochastic linear partial differential equations and stochastic differential delay equations are given for illustration.

Optimal filtering and filter stability of linear stochastic delay systems

Optimal filtering equations are obtained for very general linear stochastic delay systems. Stability of the optimal filter is studied in the case where there are no delays in the observations. Using the duality between linear filtering and control, asymptotic stability of the optimal filter is proved. Finally, the cascade of the optimal filter and the deterministic optimal quadratic control system is shown to be asymptotically stable as well.

Stability of linear stochastic differential delay systems

Sufficient conditions for stability with probability one are developed for a class of linear stochastic differential delay systems. The conditions take a form that is readily computable.