*(English)*Zbl 1025.60028

The authors consider an $n$-dimensional neutral stochastic differential delay equation with Markovian switching of the form

Here $B\left(t\right)$ is an $m$-dimensional Brownian motion. The process $r\left(t\right)$ is a right-continuous Markov chain taking values in a finite state space $S$, adapted to the filtration of the problem, but independent of the Brownian motion. The functions $D:{R}^{n}\times S\to {R}^{n}$, $f:{R}^{n}\times {R}^{n}\times {R}_{+}\times S\to {R}^{n}$ and $g:{R}^{n}\times {R}^{n}\times {R}_{+}\times S\to {R}^{n}$ are Borel-measurable functions. The authors prove the existence and uniqueness of a solution, as well as its ${L}^{p}$-boundedness on finite time intervals, under linear growth and local Lipschitz conditions. They also establish asymptotical boundedness and exponential stability in the $p$th mean, as well as almost sure exponential stability of the zero solution of (1) under sufficient conditions via a Lyapunov-type theory. In the last section several examples and their asymptotic behaviour are discussed.

##### MSC:

60H20 | Stochastic integral equations |

34K50 | Stochastic functional-differential equations |

93E15 | Stochastic stability |

93D05 | Lyapunov and other classical stabilities of control systems |