On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

Abstract

In this paper, under a local Lipschitz condition and a monotonicity condition, the problems on the existence and uniqueness theorem as well as the almost surely asymptotic behavior for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are discussed by using the Lyapunov function and some stochastic analysis techniques. Two integral lemmas are firstly established to overcome the difficulty stemming from the coexistence of the stochastic perturbation and the time-varying delay. Then, without any redundant restrictive condition on the time-varying delay, by utilizing the integral inequality, the exponential stability in pth(p ≥ 1)-moment for such equations is investigated. By employing the nonnegative semi-martingale convergence theorem, the almost sure exponential stability is analyzed. Finally, two examples are given to show the usefulness of the results obtained.