×

Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay. (English) Zbl 1195.34123

The authors prove an existence and uniqueness result for a neutral stochastic delay differential equation with unbounded delay driven by Brownian motion with initial condition in the state space of bounded continuous functions on \((-\infty,0]\) taking values in \({\mathbb R}^d\). Their basic assumption is a local Lipschitz and a linear growth condition on the coefficients with respect to the supremum norm on the state space. The existence proof is based on the usual successive approximation procedure.

MSC:

34K50 Stochastic functional-differential equations
34K05 General theory of functional-differential equations
34K40 Neutral functional-differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.; X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. · Zbl 0892.60057
[2] Friedman, A., Stochastic Differential Equations and Their Applications, vol. 2 (1976), Academic Press: Academic Press San Diego · Zbl 0323.60057
[3] Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0787.34002
[4] R.Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff and Noorhoff, Rockville, MD, 1981.; R.Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff and Noorhoff, Rockville, MD, 1981.
[5] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland Amsterdam · Zbl 0495.60005
[6] Liu, K.; Xia, X., On the exponential stability in mean square of neutral stochastic functional differential equations, Systems and Control Letters, 37, 207-215 (1999) · Zbl 0948.93060
[7] Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: Wiley New York
[8] V.B. Kolmanoskii, V.R. Nosov, Stability and periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981.; V.B. Kolmanoskii, V.R. Nosov, Stability and periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981.
[9] Mao, X., Exponential stability in mean square of neutral stochastic differential functional equations, Systems and Control Letters, 26, 245-251 (1995) · Zbl 0877.93133
[10] Luo, Q.; Mao, X.; Shen, Y., Newcriteria on exponential stability of neutral stochastic differential delay equations, Systems and Control Letters, 55, 826-834 (2006) · Zbl 1100.93048
[11] Koimanovskii, V. B., On the stability of stochastic systems with delay, Problems Information Transmission, 5, 4, 59-67 (1969) · Zbl 0279.93049
[12] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press New York · Zbl 0593.34070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.