On The Asymptotic Behavior Of Highly Nonlinear Hybrid Stochastic Delay Differential Equations

This degree's dissertation mainly considers some problems on the existence and uniqueness as well as stability for the global solution of nonlinear highly stochastic delay differential equations with Markovian switching.In order to overcome the difficulty stemming from the time-varying delay with the existence of its derivative and its derivative value being less than one,by using the Lyapunov function,some technical lemmas are firstly established;then,by using the theory of stochastic analysis,the existence and uniqueness for the global solution of such equations is proved.By utilizing the delay integral inequality and the integral lemma?Barbalat lemma and the nonnegative semi-martingale convergence theorem,the exponential stability in moment and the almost surely exponential stability for such equations are investigated,respectively.In addition,by using the integral lemma and the theory of stochastic analysis,the almost surely asymptotic stability is also analyzed.The concrete content of this dissertation is presented as follows:In the first Chapter,the background,the significance,the main innovations and some preliminary knowledge of this dissertation are successively introduced.In the second Chapter,when the locally Lipschitz condition and the generalized monotonicity condition holds for the drift term and the diffusion term,some results on the existence and uniqueness,the exponential stability in moment,the almost surely exponential stability and the almost surely asymptotic stability for the global solution of nonlinear highly stochastic delay differential equations with Markovian switching are proved.In the third Chapter,under the locally Lipschitz and the generalized monotonicity condition that are satisfied for the drift term and the diffusion term,the existence and uniqueness,the exponential stability in moment,the almost surely exponential stability and the almost surely asymptotic stability for the global solution of nonlinear highly nonautonomous stochastic delay differential equations with Markovian switching are considered.In the fourth Chapter,under the locally Lipschitz and the generalized monotonicity condition that are satisfied for the drift term and the diffusion term,the existence and uniqueness,the stability in moment,the almost surely stability for the global solution of nonlinear highly stochastic delay differential equations with Markovian switching are analyzed.It is mentioned that the involved stability analysis has a general decay rate.Stability with a general decay rate includes the exponential stability,the polynomial stability and the logarithmic one as three special cases.In the fifth Chapter,main research results on stability and decay rate are summarized,and the future research directions are given.