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Neutral stochastic differential delay equations with Markovian switching. (English) Zbl 1025.60028

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

$d\left[x\left(t\right)-D\left(x\left(t-\tau \right),r\left(t\right)\right)\right]=f\left(x\left(t\right),x\left(t-\tau \right),t,r\left(t\right)\right)dt+g\left(x\left(t\right),x\left(t-\tau \right),t,r\left(t\right)\right)dB\left(t\right)·\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 $D:{R}^{n}×S\to {R}^{n}$, $f:{R}^{n}×{R}^{n}×{R}_{+}×S\to {R}^{n}$ and $g:{R}^{n}×{R}^{n}×{R}_{+}×S\to {R}^{n}$ 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