Stability Of Stochastic Differential Delay Equations With Jumps And Markovian Switching

Since the theory of stochastic diferential equations (SDEs) was established, SDEshave been widely used in many fields. However, in the realistic world, systems may de-pend on both the present state and the past state. Systems also can have instantaneouschanges or switch among diferent states due to changes of the environment. ClassicalSDEs are not applicable to model these systems. In recent years, stochastic diferen-tial delay equations (SDDEs) with Poisson jumps and Markovian switching have beenintroduced to deal with these situations.In the study of stochastic dynamical systems, one of emphases is the analysis ofstability. This dissertation will focus on the stability analysis of SDDEs with Poissonjumps and Markovian switching.The main contributions of this dissertation are as follows:(1) We give some criteria on the mean square exponential stability of SDDEs with Pois-son jumps and Markovian switching by using Lyapunov function methods. We alsopresent and prove a theorem which indicates the mean square exponential stabilitycan imply the almost sure exponential stability under some conditions.(2) We firstly apply theψγstability to SDDEs with Poisson jumps and Markovianswitching and give some criteria on the p-th momentψγstability and the almostsureψγstability.(3) We study the robust stability of SDDEs with Poisson jumps and Markovian switch-ing. In the situation of linear uncertain perturbation, we give conditions whichshould be satisfied in order to keep the system mean square exponentially stable. Inthe situation of semi-linear uncertain perturbation, we present sufcient conditionsof the mean square exponential stability and the almost sure exponential stability.(4) We discuss the mean square exponential stability of stochastic time-varying delayinterval systems with Poisson jumps and Markovian switching. Some criteria onthe mean square exponential stability of stochastic delay interval systems are statedby using M-matrix theories and notations.(5) We study SDDEs with Poisson jumps and Markovian switching but without thelinear growth condition and prove an existence and uniqueness theorem of the so- lution. Moreover, we show that the solution is p-th moment asymptotically bound-ed. Under a few additional conditions, we prove that SDDEs with Poisson jumpsand Markovian switching but without the linear growth condition are not only p-thmoment exponentially stable but also almost surely exponentially stable.