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Stability in distribution of neutral stochastic differential delay equations with Markovian switching. (English) Zbl 1175.34103
Summary: We focus on stochastic delay differential equations in the form: for \(\tau>0\)
\[ d[x(t)-G(x(t-\tau))]=f(x(t),x(t-\tau),r(t))dt+g(x(t),x(t-\tau),r(t))dB(t),\quad t\geq 0,\tag{2.1} \]
We are concerned with neutral stochastic differential delay equations with Markovian switching (NSDDEwMSs). We derive sufficient conditions for stability in distribution and generalize some results of Basak et al. and Yuan et al. to cover a class of much more general NSDDEwMSs. In the end, two examples are established to demonstrate the theory of our work.

MSC:
34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K20 Stability theory of functional-differential equations
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[1] Anderson, W.J., Continuous-time Markov chains, (1991), Springer Berlin · Zbl 0721.60081
[2] Basak, G.K.; Bisi, A.; Ghosh, M.K., Stability of a random diffusion with linear drift, J. math. anal. appl., 202, 604-622, (1996) · Zbl 0856.93102
[3] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer New York
[4] Kolmanovskii, V.B.; Nosov, V.R., Stability and periodic modes of control systems with after effect, (1981), Nauka Moscow
[5] Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., Neutral stochastic differential delay equations with Markovian switching, Stoch. anal. appl., 21, 819-847, (2003) · Zbl 1025.60028
[6] Luo, J., Comparison principle and stability of ito stochastic differential delay equations with Poisson jump and Markovian switching, Nonlinear anal., 64, 253-262, (2006) · Zbl 1082.60054
[7] Luo, J.; Zou, J.; Hou, Z., Comparison principle and stability criteria for stochastic delay differential equations with Markovian switching, Sci. China, 46, 129-138, (2003) · Zbl 1217.60046
[8] Mao, X., Stochastic differential equations and applications, (1997), Horwood NewYork · Zbl 0874.60050
[9] Mao, X.; Matasov, A.; Piunovskiy, A.B., Stochastic differential delay equations with Markovian switching, Bernoulli, 6, 73-90, (2000) · Zbl 0956.60060
[10] Mao, X.; Shen, Y.; Yuan, C., Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stoch. proc. appl., 118, 1385-1406, (2008) · Zbl 1143.60041
[11] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperical College Press · Zbl 1126.60002
[12] Mohammed, S.-E.A., Stochastic functional differential equations, (1984), Longman Scientific and Technical · Zbl 0584.60066
[13] Øksendal, B., Stochastic differential equations, (2003), Springer
[14] Yuan, C.; Mao, X., Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica, 40, 343-354, (2004) · Zbl 1040.93069
[15] Yuan, C.; Zou, J.; Mao, X., Stability in distribution of stochastic differential delay equations with Markovian switching, Syst. control lett., 50, 195-207, (2003) · Zbl 1157.60330
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