In this note, the distributed consensus corrupted by relativestate-dependent measurement noises is considered. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a vector function of agents' relative states. By investigating the structure of this interaction and the tools of stochastic differential equations, we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain, the number of agents and the noise intensity function to ensure mean square (m. s.) and almost sure (a. s.) consensus and quantify the convergence rate and the steadystate error. Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure m. s. consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions.

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma

This paper establishes Razumikhin-type theorems on general decay stability for stochastic functional differential equations. This improves existing stochastic Razumikhin-type theorems and can make us examine the stability with general decay rate in the sense of the pth moment and almost sure. These stabilities may be specialized as the exponential stability and the polynomial stability. When the almost sure stability is examined, the conditions of this paper may defy the linear growth condition for the drift term, which implies that the theorems of this paper can work for some cases to which the existing results cannot be applied. This paper also examines some sufficient criteria under which this stability is robust. To illustrate applications of our results clearly, this paper also gives two examples and examines the exponential stability and the polynomial stability, respectively.By condition (H2), letting u = /( +¯ ) and applying the inequality (40) yield (here we assume that ,¯ = 0, otherwise this computation is easier) |g(t, )| 2 1 (t) ( +¯ )| (0)| 2 +¯ ( +¯ ) 0 − | ( )| 2 d ( ) .

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In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation _ xðtÞ ¼ f ðxðtÞ, tÞ, in which f satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation dxðtÞ ¼ f ðxðtÞ, tÞdt þ qxðtÞdw 1 ðtÞ þ jxðtÞj xðtÞdw 2 ðtÞ. This article shows that appropriate may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable.

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Abstract. This paper examines the numerical solutions of neutral stochastic functional differential equations (NSFDEs)The key contribution is to establish the strong mean square convergence theory of the EulerMaruyama approximate solution under the local Lipschitz condition, the linear growth condition and the contractive mapping. These conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here are obtained under quite general conditions. Although the way of analysis borrows from [16], to cope with u(xt), several new techniques have been developed.

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