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**A note on the LaSalle-type theorems for stochastic differential delay equations.**
*(English)*
Zbl 0996.60064

The author presents an improvement of results obtained in an earlier paper [J. Math. Anal. Appl. 236, No. 2, 350-369 (1999; Zbl 0958.60057)]. In the article \(n\)-dimensional stochastic differential delay equations are considered, which are of the form
\[
dx(t)=f(x(t),x(t-\tau),t) dt + g(x(t),x(t-\tau),t) dB(t),\tag{1}
\]
where \(B(t)\) denotes \(m\)-dimensional Brownian motion. The main theorem is a stochastic version of the LaSalle theorem, providing criteria for the determination of the almost sure asymptotic behaviour of the solution of (1). The improvement concerns the assumptions on the coefficient functions. The local Lipschitz and local linear growth conditions on \(f\) and \(g\) are relaxed to local boundedness in the first two arguments and uniform boundedness in the last argument of \(f\) and \(g\), in addition the existence and uniqueness of a solution of (1) is required. The results can thus be applied to a larger class of equations. The proof of the theorem, some corollaries and an extension to the multiple delay case are given. Several examples are presented, demonstrating the usefulness of the results.

Reviewer: Evelyn Buckwar (Berlin)

### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34K50 | Stochastic functional-differential equations |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

### Keywords:

stochastic delay differential equations; stochastic stability; Lyapunov stability; LaSalle theorem### Citations:

Zbl 0958.60057### References:

[1] | Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0787.34002 |

[2] | LaSalle, J. P., Stability theory of ordinary differential equations, J. Differential Equations, 4, 57-65 (1968) · Zbl 0159.12002 |

[3] | R. Sh. Liptser, and, A. N. Shiryayev, Theory of Martingales, Kluwer Academic, Dordrecht, 1989.; R. Sh. Liptser, and, A. N. Shiryayev, Theory of Martingales, Kluwer Academic, Dordrecht, 1989. · Zbl 0728.60048 |

[4] | Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050 |

[5] | Mao, X., LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236, 350-369 (1999) · Zbl 0958.60057 |

[6] | Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Dekker: Dekker New York · Zbl 0851.93074 |

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