ON THE GENERAL DECAY STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

Title & Authors
ON THE GENERAL DECAY STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY
Meng, Xuejing; Yin, Baojian;

Abstract
This work focuses on the general decay stability of nonlinear stochastic differential equations with unbounded delay. A Razumikhin-type theorem is first established to obtain the moment stability but without almost sure stability. Then an improved edition is presented to derive not only the moment stability but also the almost sure stability, while existing Razumikhin-type theorems aim at only the moment stability. By virtue of the $\small{M}$-matrix techniques, we further develop the aforementioned Razumikhin-type theorems to be easily implementable. Two examples are given for illustration.
Keywords
stochastic delay differential equation;unbounded delay;Razumikhin-type theorem;stability;M-matrix;
Language
English
Cited by
1.
The Razumikhin approach on general decay stability for neutral stochastic functional differential equations, Journal of the Franklin Institute, 2013, 350, 8, 2124
References
1.
J. A. D Appleby, Decay and growth rates of solutions of scalar stochastic delay differential equations with unbounded delay and state dependent noise, Stoch. Dyn. 5 (2005), no. 2, 133-147.

2.
J. A. D. Appleby and E. Buckwar, Sufficient condition for polynomial asymp- totic behavier of stochastic pantograph equations, available at www.dcu.ie/maths/research/preprint.shtml.

3.
L. Arnold, Stochastic Differential Equations: Theory and applications, Wiley, New York, 1974.

4.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA. 1994.

5.
J. Cermak, The asymptotics of solutions for a class of delay differential equations, Rocky Mountain J. Math. 33 (2003), no. 3, 775-786.

6.
J. Cermak and Stanislava Dvorakova, Asymptotic estimation for some nonlinear delay differential equations, Recults Math. 51 (2008), no. 3-4, 201-213.

7.
M. L. Heard, A change of variables for functional differential equations, J. Differential Equations 18 (1975), 1-10.

8.
D. J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), no. 2, 592-609.

9.
A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. 4 (1993), no. 1, 1-38.

10.
R. Z. Khas'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1981.

11.
V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.

12.
G. Makay and J. Terjeki, On the asymptotic behavior of the pantograph equations, Elec- tron. J. Qual. Theory Differ. Equ. 1998 (1998), no. 2, 1-12.

13.
X. Mao, Exponential Stability of Stochastic Differential Equations, Dekker, 1994.

14.
X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process, Stochastic Process. Appl. 65 (1996), no. 2, 233- 250.

15.
X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28 (1997), no. 2, 389-401.

16.
X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.

17.
X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl. 268 (2002), no. 1, 125-142.

18.
X. Mao, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl. 236 (1999), no. 2, 350-369.

19.
X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Non- linear Stud. 7 (2000), no. 2, 307-328.

20.
X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and technical, 1991.

21.
X. Mao, Stability and stabilisation of stochastic differential delay equations, IET Control Theory Appl. 1 (2007), no. 6, 1551-1566.

22.
X. Mao, A. Matasov, and A. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli 6 (2000), no. 1, 73-90.

23.
X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl. 23 (2005), no. 5, 1045-1069.

24.
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College press, 2006.

25.
Y. Shen, Q. Luo, and X. Mao, The improved LaSalle-type theorems for stochastic func- tional differential equations, J. Math. Anal. Appl. 318 (2006), no. 1, 134-154.