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Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations. (English) Zbl 1166.60040
Summary: The paper discusses both $p$th moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations, by using the Razumikhin-type technique. The main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with Lyapunov functionals. The analysis is based on a paper of X. Mao [SIAM J. Math. Anal. 28, No. 2, 389–401 (1997; Zbl 0876.60047)], referring to mean square and almost sure exponential stability.
##### MSC:
 60H20 Stochastic integral equations 34K50 Stochastic functional-differential equations
##### References:
 [1] Haddock, J. R.; Krisztin, T.; Terjéki, J.; Wu, J. H.: An invariance principle of Lyapunov – razumikhin type for neutral functional differential equations, J. differential equations 107, 395-417 (1994) · Zbl 0796.34067 · doi:10.1006/jdeq.1994.1019 [2] Hale, J. K.; Meyer, K. R.: A class of functional equations of neutral type, Mem. amer. Math. soc. 76, 1-65 (1967) · Zbl 0179.20501 [3] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1991) [4] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) [5] Janković, S.; Jovanović, M.: The p-th moment exponential stability of neutral stochastic functional differential equations, Filomat 20, No. 1, 59-72 (2006) · Zbl 1142.60371 · doi:10.2298/FIL0601059J [6] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional differential equations, (1986) [7] Kolmanovskii, V. B.; Myshkis, A.: Applied theory of functional differential equations, (1992) [8] Liao, X. X.; Mao, X.: Almost sure exponential stability of neutral differential difference equations with damped stochastic perturbations, Electron. J. Probab. 1, No. 8, 1-16 (1986) · Zbl 0891.60051 · doi:emis:journals/EJP-ECP/EjpVol1/paper8.abs.html [9] Liu, K.; Mao, X.: Exponential stability of non-linear stochastic evolution equations, Stochastic process. Appl. 78, 173-193 (1998) · Zbl 0933.60072 · doi:10.1016/S0304-4149(98)00048-9 [10] Luo, Z.; Shen, J.: New razumikhin-type theorems for impulsive functional differential equations, Appl. math. Comput. 125, 375-386 (2002) · Zbl 1030.34078 · doi:10.1016/S0096-3003(00)00139-9 [11] Mao, X.: Exponential stability of stochastic differential equations, (1994) [12] Mao, X.: Exponential stability in mean square of neutral stochastic differential functional equations, Systems control lett. 26, 245-251 (1995) · Zbl 0877.93133 · doi:10.1016/0167-6911(95)00018-5 [13] Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic process. Anal. 65, 233-250 (1996) · Zbl 0889.60062 · doi:10.1016/S0304-4149(96)00109-3 [14] Mao, X.: Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. anal. 28, No. 2, 389-401 (1997) · Zbl 0876.60047 · doi:10.1137/S0036141095290835 [15] Mao, X.: Stochastic differential equations and applications, (1997) [16] Mohammed, E. A.: Stochastic functional differential equations, (1986) [17] Randjelović, J.; Janković, S.: On the p-th moment exponential stability criteria of neutral stochastic functional differential equations, J. math. Anal. appl. 326, 266-280 (2007) · Zbl 1115.60065 · doi:10.1016/j.jmaa.2006.02.030 [18] Razumikhin, B. S.: On the stability of systems with a delay, Prikl. mat. Mekh. 20, 500-512 (1956) [19] Razumikhin, B. S.: Application of Lyapunov’s method to problems in the stability of systems with a delay, Avtomat. i telemekh. 21, 740-749 (1960) · Zbl 0114.04502