Stability analysis of neutral stochastic differential delay equations driven by Lxc3xa9vy noises Abstract   This paper mainly analyzes the wellposedness, and the stability analysis for the global solution of neutral stochastic differential delay equations (NSDDEs) driven by Levy noises. By using an integral lemma and a Lyapunov function approach, the existence and uniqueness theorem is proved. Then, by using the inequality technique and the stochastic analysis theory, the exponential stability in pth(pxe2x80xafxe2x89xa5xe2x80xaf2) moment of such equations is discussed. By using another integral lemma, and using the Baralat lemma as well as the stochastic analysis, the almost surely asymptotic stability is also studied. Finally, one example is given to check the effectiveness of the findings derived.

On local stability of stochastic delay nonlinear discrete systems with statedependent noise Abstract   We examine the local stability of solutions of a delay stochastic nonlinear difference equation with deterministic and statedependent Gaussian perturbations. We apply the degenerate Lyapunovxe2x80x93Krasovskii functional technique and construct a sequence of events, each term of which is defined by a bound on a normally distributed random variable. Local stability holds on the intersection of these events, which has probability at least     1  xe2x88x92  xcexb3  ,     xcexb3xe2x80xafxe2x88x88xe2x80xaf(0, 1). This probability can be made arbitrarily high by choosing the initial value sufficiently small. We also present a generalization to systems where a condition for stability is expressed in terms of the diagonal part of the unperturbed system, and computer simulations which illustrate our results.