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Comparison principle and stability of Itô stochastic differential delay equations with Poisson jump and Markovian switching. (English) Zbl 1082.60054
Summary: The comparison principle for the nonlinear Itô stochastic differential delay equations with Poisson jump and Markovian switching is established. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in the $p$th mean, asymptotic stability in the $p$th mean and the $p$th moment exponential stability of such equations. Some known results are generalized and improved.

##### MSC:
 60H10 Stochastic ordinary differential equations 34K50 Stochastic functional-differential equations
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##### References:
 [1] 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 [2] Gikhman, I. I.; Skorohod, A. V.: Stochastic differential equations. (1972) · Zbl 0242.60003 [3] Ikeda, N.; Watanabe, S.: Stochastic differential equations and diffusion processes. (1989) · Zbl 0684.60040 [4] Ji, Y.; Chizeck, H. J.: Controllability, stability and continuous-time Markovian jump linear quadratic control. IEEE trans. Automat. control 35, 777-788 (1990) · Zbl 0714.93060 [5] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, vol. II, Academic Press, New York, 1969. · Zbl 0177.12403 [6] Luo, J.; Zou, J.; Hou, Z.: Comparison principle and stability criteria for stochastic delay differential equations with Markovian switching. Sci. China 46, No. 1, 129-138 (2003) · Zbl 1217.60046 [7] Mao, X.: Stability of stochastic differential equations with Markovian switching. Stochastic processes appl. 79, 45-67 (1999) · Zbl 0962.60043 [8] Mao, X.: Stochastic functional differential equations with Markovian switching. Funct. differential equations 6, 375-396 (1999) · Zbl 1034.60063 [9] Mao, X.: Robustness of stability of stochastic differential delay equations with Markovian switching. Stability control: theory appl. 3, No. 1, 48-61 (2000) [10] Mao, X.: Asymptotic stability for stochastic differential delay equations with Markovian switching. Funct. differential equations 9, No. 1 -- 2, 201-220 (2002) · Zbl 1095.34556 [11] Mao, X.: Exponential stability of stochastic delay interval systems with Markovian switching. IEEE trans. Automat. control 47, No. 10, 1604-1612 (2002) [12] Mao, X.: Asymptotic stability for stochastic differential equations with Markovian switching. WSEAS trans. Circuits 1, No. 1, 68-73 (2002) · Zbl 1095.34556 [13] Mao, X.; Matasov, A.; Piunovskiy, A. B.: Stochastic differential delay equations with Markovian switching. Bernoulli 6, No. 1, 73-90 (2000) · Zbl 0956.60060 [14] Mao, X.; Shaikhet, L.: Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching. Stability control: theory appl. 3, No. 2, 87-101 (2000) [15] Mariton, M.: Jump linear systems in automatic control. (1990) [16] Skorohod, A. V.: Asymptotic methods in the theory of stochastic differential equations. (1989) [17] Svishchuk, A. V.; Kazmerchuk, Yu.I.: Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance. Theor. probab. Math. stat. 64, 167-178 (2002) [18] Yuan, C.; Mao, X.: Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stochastic processes appl. 103, 277-291 (2003) · Zbl 1075.60541