Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations. (English) Zbl 1232.60046

The authors derive new criteria for asymptotic stability and boundedness of solutions of stochastic functional differential equations, whereby the linear growth condition is replaced by a more general condition.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
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[1] Appleby, J.A.D.; Reynolds, D.W., Decay rates of solutions of linear stochastic Volterra equations, Electron. J. probab., 13, 922-943, (2008) · Zbl 1188.45008
[2] Appleby, J.A.D.; Rodkina, A., Stability of nonlinear stochastic Volterra difference equations with respect to a fading perturbation, Int. J. difference equ., 4, 165-184, (2009)
[3] Bahar, A.; Mao, X., Persistence of stochastic power law logistic model, J. appl. probab. stat., 3, 1, 37-43, (2008) · Zbl 1302.60090
[4] Basin, M.; Rodkina, A., On delay-dependent stability for a class of nonlinear stochastic systems with multiple state delays, Nonlinear anal., 68, 2147-2157, (2008) · Zbl 1154.34044
[5] Khasminskii, R.Z., (), Alphen: Sijtjoff Noordhoff (translation of the Russian edition)
[6] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers · Zbl 0917.34001
[7] Kolmanovskii, V.B.; Nosov, V.R., Stability and periodic modes of control systems with aftereffect, (1981), Nauka Moscow · Zbl 0457.93002
[8] Lipster, R.Sh.; Shiryayev, A.N., Theory of martingales, (1989), Kluwer Academic Publisher · Zbl 0728.60048
[9] Loève, M., Probability theory, (1963), D. Van Nostrand Company Canada · Zbl 0108.14202
[10] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Scientific and Technical · Zbl 0724.60059
[11] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker · Zbl 0851.93074
[12] Mao, X., Exponential stability in Mean square of neutral stochastic differential functional equations, Syst. control lett., 26, 245-251, (1995) · Zbl 0877.93133
[13] Mao, X., Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. math. anal., 28, 2, 389-401, (1997) · Zbl 0876.60047
[14] Mao, X., Stochastic differential equations and applications, (2007), Horwood Publishing Chichester
[15] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002
[16] Mohammed, S.-E.A., Stochastic functional differential equations, (1986), Longman Scientific and Technical · Zbl 0584.60066
[17] Natanson, I.P., Theory of functions of real variables, vol. 1, (1964), Frederick Unger Pub. Co New York · Zbl 0133.31101
[18] Rodkina, A.; Basin, M., On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term, Syst. control lett., 56, 423-430, (2007) · Zbl 1124.93066
[19] Shu, Z.; Lam, J.; Xu, S., Improved exponential estimates for neutral systems, Asian J. control, 11, 261-270, (2009)
[20] Yang, R.; Gao, H.; Lam, J.; Shi, P., New stability criteria for neural networks with distributed and probabilistic delays, Circuits syst. signal process, 28, 505-522, (2009) · Zbl 1170.93027
[21] Yuan, C.; Lygeros, J., Stabilization of a class of stochastic differential equations with Markovian switching, Syst. control lett., 54, 819-833, (2005) · Zbl 1129.93517
[22] Yuan, C.; Lygeros, J., Asymptotic stability and boundedness of delay switching diffusions, IEEE trans. automat. control, 51, 171-175, (2006) · Zbl 1366.93453
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