Study On The Stability Of Solutions Of Several Hybrid Stochastic Unbounded Delay Differential Equations

With the increasing attention people pay to the influence of random factors and time delays on power system,stochastic delay differential equations have been widely used in many areas,such as engineering mechanics,biotechnology and finance,etc.The internal or external changes of the system may lead to random changes that make system switch between finite states,such as the microbial cultivation and network con trol system.Continuous time Markov chains can be used to model these changes.Besides,the abrupt changes of the system,for example severe shock or collapse in financial market,can be described by the jump processes.Stability is central to the dynamical systems.The stability theory research of stochastic differential equations is an important aspect of the study of qualitative theory of its solutions.At the same time,unbounded time delay is also widely used in ecology,economic problems and power systems.So it is full of theoretical and practical value to study the stability of the solutions of hybrid stochastic differential equations with unbounded delay.In this thesis,we study the stability of three kinds of hybrid stochastic differen tial equations with unbounded delay.Firstly,the stability of the solution of neutral stochastic unbounded delay differential equations with Markovian switching is investi gated.We obtain the existence and uniqueness,the boundedness and the criteria of pth moment exponential stability and almost sure exponential stability of its solutions.Sec ondly,we study the stability of the solution of stochastic unbounded delay differential equations with Poisson jump and Markovian switching,and we obtain the boundedness and the criteria of pth moment exponential stability and almost sure exponential stabil ity of its solutions.Thirdly,we extend the model to corresponding neutral case,and study the stability of the solution of hybrid neutral stochastic unbounded delay differ ential equations with Poisson jumps.We obtain the criteria of pth moment exponential stability and almost sure exponential stability of the its solutions.For each model,the validity of the results is illustrated by simulating numerical examples.The Lyapunov functions method,generalized Ito formula and nonnegative semi martingale convergence theorem are used in this thesis.The factor e-??(t)is used to overcome the difficulties caused by unbounded delay.One of the contributions of this thesis is improving the bounded delay of stochastic delay differential equations to the general unbounded case.The other one is studying the hybrid stochastic differential e quations with conjunction of unbounded delay,Markovian switching,neutral term and Poisson jumps.