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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 $$d[x(t) - D(x(t-\tau),r(t))] = f(x(t),x(t-\tau),t,r(t)) dt + g(x(t),x(t-\tau),t,r(t)) dB(t).\tag 1$$ Here $B(t)$ is an $m$-dimensional Brownian motion. The process $r(t)$ 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 $D:R^n\times S \rightarrow R^n$, $f:R^n \times R^n \times R_+ \times S \rightarrow R^n$ and $g:R^n \times R^n \times R_+ \times S \rightarrow R^n$ 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.

60H20Stochastic integral equations
34K50Stochastic functional-differential equations
93E15Stochastic stability
93D05Lyapunov and other classical stabilities of control systems
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