Asymptotic Behavior Of Stochastic Differential Systems With Mixed Delays

A real dynamic system is influenced by both stochastic disturbances and time delays, so when we consider the behavior of a dynamic system, we use the stochastic differential delay equation and the stochastic functional differential equation as mod-eling tools to investigate asymptotic properties of stochastic dynamic systems with discrete delays or distributed delays. So far, these topics have been received a lot of attention and there are so many references about them. However, discrete delays and distributed delays coexist in real dynamic systems, thus, it is reasonable to consider them together and it leads us to investigate stochastic differential systems with mixed delays.Stochastic differential systems with mixed delays are received increasing atten-tion recently, but systematic research has not been developed yet. This doctoral dis-sertation fill the gap in a sense. In this dissertation, we first consider the existence and uniqueness of the global solution and then by employing the Borel-Cantelli Lemma, the semi-martingale convergence theorem, the exponential martingale inequality and so on, the boundedness, stability and pathwise growth properties of stochastic differ-ential systems with mixed delays are considered. Precisely speaking, the pathwise growth property here means there is an upper limit that paths grow no faster than it and also we care about the speed value is acceptable or not. In comparison with known works, this dissertation allows coefficients of equations grow polynomially.The last three sections discuss the Kolmogorov system, Lotka-Volterra system and neural networks. Undoubtedly, stochastic Kolmogorov models and stochastic Lotka-Volterra models play an important role in ecology which are used to model the possible changes of biological population and construction caused by the influence of natural environment and something else. In this dissertation, we consider the non-explosion,moment boundedness and pathwise growth of the two models. The neural networks model was established by Hopfield in the year 1982. After tens of years, the neural networks are not only the theoretical models of feedback dynamic systems but also useful important tools to parallel computing. During the procession of com-puting, there exist a lot of stochastic disturbances. So it is important to know how stochastic disturbances affect neural networks, especially, we wonder if the networks will lose or keep stability with those disturbances. In this dissertation, by controlling the growth of equations's coefficients, we obtain our desired properties and all the criteria are showed in the form of inequali-ties or matrices. Different with known works, by employing the Lyapunov function (?), M-matrix technique are used frequently. This method guarantees that the obtained criteria are easy to be verified since no undetermined parameters are included. Moreover, because the mixed delay models include some discrete delay models and distributed delay models as special cases, our results cover and extend some existing conclusions on to stochastic differential delay equations and stochastic functional differential equations.