: In this paper we study both pth moment and almost sure exponential stability of the stochastic differential delay equation dx(t)=f(t;x(t);x(t\Gammaø))dt+g(t;x(t);x(t\Gammaø))dw(t). Introduce the corresponding stochastic differential equation (without delay) dx(t)=f(t;x(t);x(t))dt+ g(t;x(t);x(t))dw(t) and assume it is exponentially stable which is guaranteed by the existence of the Lyapunov function. We shall show that the original stochastic differential delay equation remains exponentially stable provided the time lag ø is sufficiently small, and a bound for such ø is obtained. Key Words: stochastic differential delay equations, Lyapunov function, Lyapunov exponent, Borel-Cantelli lemma. AMS 1991 Classifications: 60H10, 34K30 1. Introduction In many branches of science and industry stochastic differential delay equations have been used to model the evolution phenomena because the measurements of timeinvolving variables and their dynamics usually contain some delays (cf. Kolmanov...

Abstract. Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler–Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p>2. As an application of this general theory we show that an implicit variant of Euler–Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler–Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

Although deterministic interval systems have received a great deal of attention, so far there is no work on stochastic interval systems. The main aim of this paper is to initiate the study of stochastic interval systems. Of course there are many properties of such systems to be investigated, but this paper will concentrate on the study of exponential stability of stochastic interval systems with time-varying delays. The main technique used in this paper is the Razumikhin-type theorem established recently by Mao [11]. Key Words: stochastic interval system, Razumikhin-type theorem, Brownian motion, Ito's formula.

In this paper we discuss stochastic differential delay equations with MarkovJan switching. Such an equation can be regarded as the result of several stochastic differential delay equations switching from one to the others according to the movement of a Markov chain. One of the main aims of this paper is to investigate the exponential stability of the equations.

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is one-sided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Euler–Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p> 2. Under our assumptions, both SSBE and CSSBE give well-defined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential mean-square stability in the case where the drift has a negative one-sided Lipschitz constant. This work extends the deterministic nonlinear stability the-

The main aim of this paper is to establish stochastic versions of the well-known LaSalle stability theorem. From these stochastic versions follow many classical results on stochastic stability. This shows clearly the power of our new results.

. We present the first result on global outputfeedback stabilization (in probability) for stochastic nonlinear continuous-time systems. The class of systems that we consider is a stochastic counterpart of the broadest class of deterministic systems for which globally stabilizing controllers are currently available. Our controllers are "inverse optimal" and possess an infinite gain margin. A reader of the paper needs no prior familiarity with techniques of stochastic control. 1 Introduction Despite huge popularity of the LQG control problem, the stabilization problem for nonlinear stochastic systems has hardly received any attention until recently. Efforts toward (global) stabilization of stochastic nonlinear systems have been initiated in the work of Florchinger [5, 6, 7] who, among other things, extended the concept of control Lyapunov functions and Sontag's stabilization formula [22] to the stochastic setting. A breakthrough towards arriving at constructive methods for stabilizati...

Abstract. A class of implicit methods is introduced for Ito stochastic differential equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same strong finite-time convergence rate as the explicit Euler–Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue of mean-square A-stability is studied. Weak variants are also considered and their stability analyzed. Key words. A-stability, backward Euler, Euler–Maruyama, linear stability, Poisson process, stochastic differential equation, strong convergence, theta method, trapezoidal rule. AMS subject classifications. 65C30, 65L20, 60H10

Hopfield (1984 Proc. Natl Acad. Sci. USA 81 3088–92) showed that the time evolution of a symmetric neural network is a motion in state space that seeks out minima in the system energy (i.e. the limit set of the system). In practice, aneuralnetwork is often subject to environmental noise. It is therefore useful and interesting to find out whether the system still approaches some limit set under stochastic perturbation. In this paper, we will give a number of useful bounds for the noise intensity under which the stochastic neural network will approach its limit set.

Abstract. Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler–Maruyama (EM) method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability.