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Existence-uniqueness and continuation theorems for stochastic functional differential equations. (English) Zbl 1161.34055
The authors investigate existence, uniqueness and continuation of solutions for stochastic functional differential equations driven by Brownian motion in which the coefficients map $[0,T) \times L^2(\Omega,C)$ to $L^2(\Omega,C)$. Here, $\Omega$ is the underlying probability space and $C=C([-r,0],{\mathbb{R}}^n)$, where $r>0$ is the maximal delay. Under suitable conditions like adaptedness and local Lipschitz conditions, they establish local existence and uniqueness of solutions. Due to the particular set-up (in $L^2$), maximal solutions are defined on a {\it deterministic} time interval. In addition, the authors provide sufficient conditions for global existence in terms of Lyapunov functions.

34K50Stochastic functional-differential equations
34K05General theory of functional-differential equations
60H10Stochastic ordinary differential equations
Full Text: DOI
[1] Arnold, L.: Stochastic differential equations: theory and applications, (1972) · Zbl 0216.45001
[2] Friedman, A.: Stochastic differential equations and applications, (1975) · Zbl 0323.60056
[3] Doob, J. L.: Martingales and one-dimensional diffusion, Trans. amer. Math. soc. 78, 168-208 (1955) · Zbl 0068.11301 · doi:10.2307/1992954
[4] Dynkin, E. B.: Markov processes, (1963) · Zbl 0132.37701
[5] Has’minskiĭ, R. Z.: Stochastic stability of differential equations, (1980)
[6] øksendal, B.: Stochastic differential equations: an introduction with applications, (1995) · Zbl 0841.60037
[7] Taniguchi, T.: On sufficient conditions for nonexplosion of solutions to stochastic differential equations, J. math. Anal. appl. 153, 549-561 (1990) · Zbl 0715.60072 · doi:10.1016/0022-247X(90)90231-4
[8] Taniguchi, T.: Successive approximations to solutions of stochastic differential equations, J. differential equations 96, 152-169 (1992) · Zbl 0744.34052 · doi:10.1016/0022-0396(92)90148-G
[9] Lu, Kening; Schmalfuss, B.: Invariant manifolds for stochastic wave equations, J. differential equations 236, 460-492 (2007) · Zbl 1113.37056 · doi:10.1016/j.jde.2006.09.024
[10] Shen, Y.; Luo, Q.; Mao, X.: The improved lasalle-type theorems for stochastic functional differential equations, J. math. Anal. appl. 318, 134-154 (2006) · Zbl 1090.60059 · doi:10.1016/j.jmaa.2005.05.026
[11] Wei, F.; Wang, K.: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. math. Anal. appl. 331, 516-531 (2007) · Zbl 1121.60064 · doi:10.1016/j.jmaa.2006.09.020
[12] Yang, Z.; Xu, D.; Xiang, L.: Exponential p-stability of impulsive stochastic differential equations with delays, Phys. lett. A 359, 129-137 (2006) · Zbl 1236.60061
[13] Liu, K.: Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions, Stochastic process. Appl. 115, 1131-1165 (2005) · Zbl 1075.60078 · doi:10.1016/j.spa.2005.02.006
[14] Taniguchi, T.; Liu, K.; Truman, A.: Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. differential equations 181, 72-91 (2002) · Zbl 1009.34074 · doi:10.1006/jdeq.2001.4073
[15] Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic process. Appl. 65, 233-250 (1996) · Zbl 0889.60062 · doi:10.1016/S0304-4149(96)00109-3
[16] Chang, M.: On razumikhin-type stability conditions for stochastic functional differential equations, Math. modelling 5, 299-307 (1984) · Zbl 0574.60065 · doi:10.1016/0270-0255(84)90007-1
[17] Mao, X.: Exponential stability of stochastic differential equations, (1994) · Zbl 0806.60044
[18] Mohammed, S. -E.A.: Stochastic functional differential equations, (1984) · Zbl 0584.60066
[19] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057
[20] Philip, H.: Ordinary differential equations, (1982) · Zbl 0476.34002
[21] Driver, R. D.: Ordinary and delay differential equations, (1977) · Zbl 0374.34001
[22] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1993) · Zbl 0787.34002
[23] Miller, Richard K.; Michel, Anthony N.: Ordinary differential equations, (1982) · Zbl 0552.34001
[24] Amemiya, T.: On the stability of non-linearly interconnected systems, Internat. J. Control 34, 513-527 (1981) · Zbl 0478.93008 · doi:10.1080/00207178108922545
[25] Wang, L.; Xu, D.: Global exponential stability of reaction -- diffusion Hopfield neural networks with time-varying delays, Sci. China ser. E 33, 488-495 (2003)
[26] Wintner, A.: The non-local existence problem of ordinary differential equations, Amer. J. Math. 67, 277-284 (1945) · Zbl 0063.08284 · doi:10.2307/2371729