Asymptotic Properties Of Solutions Of Neutral Stochastic Functional Differential Equations

It is a long time for the research about stochastic differential equation theory and there were lots of useful results. Requirements of the chemical engineering systems and the theory of aeroelasticity propel the research about the neutral stochastic functional differential equation theory. This paper discuss the asymptotic properties of several types of neutral stochastic functional differential equations.Firstly, we investigate the stability of neutral sochastic differential equations with multiple variable delays. The equations contain the general neutral sochastic differential equations with delays. By using a specific Lyapunov function we establish the moment stability criteria which are relatively easy to verify the moment stability of such equations. And we introduce a new kind ofψ-functions to our work and consider the stability which is more general and representative than the exponential stability and polynomial stability. With the help of stochastic calculous, moment inequality, Burkholder-Davis-Gundy inequality, etc., the almost sure stability is derived from the moment stability by appending the proper conditions.The existence of global solutions of equations is the precondition to consider the asymptotic properties. The linear growth condition is the most simple condition to ensure the existence of the global solutions. Ordinaryly, with local Lipschitz condition but not the linear growth condition, we can get the existence of local solutions. So it is important to find other conditions. By the special treatment of the neutral term and appending the proper conditions, we obtained the existence of global solutions of neutral stochastic functional differential equations. And with similar conditions we also obtained the conclusion including the moment boundedness and the time average moment boundedness. Moreover with the detailed growth conditions, we got the existence and correlative conclusions. Last, with the special form of the equation, we obtainded the convenient application conclusions to neutral stochastic differential equations with delays.The biggest delayτdepict the limitation of system memory to neutral sochastic differential equations with finite delays. In fact, it is difficult to determineτ. So it is necessary to consider the neutral stochastic differential equations with infinite delays. We introduce a new kind ofψ-functions to our work and consider the moment estimation and the almost sure estimation. With the monotone increasing and monotone decreasing ofψ-functions, we obtained some useful conclusions.At the end of this paper, we consider neutral stochastic functional differential equations with Markovian switching. By a special treatment of the neutral term and appending certain conditions, we obtained the existence of the global solutions of neutral stochastic functional differential equations with Markovian switching. And with similar conditions we also obtained the conclusion including the moment boundedness and the time average moment boundedness. Moreover with the detailed growth conditions, we got the existence and correlative conclusions.