Neutral stochastic differential delay equations with Markovian switching

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kohrianovskii, V.B. and Nosov, VR., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters Of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous-time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and stability.