On delay-dependent stability for a class of nonlinear stochastic delay-differential equations

Global asymptotic stability conditions for discrete nonlinear scalar stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov-Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. The obtained results are compared to some previously known asymptotic stability conditions for discrete nonlinear stochastic systems. An illustrative example is given.