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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 deterministic time interval. In addition, the authors provide sufficient conditions for global existence in terms of Lyapunov functions.

MSC:

34K50 Stochastic functional-differential equations
34K05 General theory of functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: Wiley New York
[2] Friedman, A., Stochastic Differential Equations and Applications (1975), Academic Press: Academic Press New York
[3] Doob, J. L., Martingales and one-dimensional diffusion, Trans. Amer. Math. Soc., 78, 168-208 (1955) · Zbl 0068.11301
[4] Dynkin, E. B., Markov Processes (1963), Fizmatgiz: Fizmatgiz Moscow · Zbl 0132.37701
[5] Has’minskiĭ, R. Z., Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff: Sijthoff and Noordhoff Maryland · Zbl 0276.60059
[6] Øksendal, B., Stochastic Differential Equations: An Introduction with Applications (1995), Springer-Verlag · 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
[8] Taniguchi, T., Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96, 152-169 (1992) · Zbl 0744.34052
[9] Lu, Kening; Schmalfuss, B., Invariant manifolds for stochastic wave equations, J. Differential Equations, 236, 460-492 (2007) · Zbl 1113.37056
[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
[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
[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
[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
[15] Mao, X., Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65, 233-250 (1996) · Zbl 0889.60062
[16] Chang, M., On Razumikhin-type stability conditions for stochastic functional differential equations, Math. Modelling, 5, 299-307 (1984) · Zbl 0574.60065
[17] Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Marcel Dekker: Marcel Dekker New York · Zbl 0851.93074
[18] Mohammed, S.-E. A., Stochastic Functional Differential Equations (1984), Pitman · Zbl 0584.60066
[19] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood · Zbl 0874.60050
[20] Philip, H., Ordinary Differential Equations (1982), Birkhäuser: Birkhäuser New York
[21] Driver, R. D., Ordinary and Delay Differential Equations (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0374.34001
[22] Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[23] Miller, Richard K.; Michel, Anthony N., Ordinary Differential Equations (1982), Academic Press: Academic Press New York · Zbl 0552.34001
[24] Amemiya, T., On the stability of non-linearly interconnected systems, Internat. J. Control, 34, 513-527 (1981) · Zbl 0478.93008
[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
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