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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.

MSC:

60H20 Stochastic integral equations
34K50 Stochastic functional-differential equations
93E15 Stochastic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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