# American Institute of Mathematical Sciences

November  2014, 19(9): 2963-2991. doi: 10.3934/dcdsb.2014.19.2963

## Second moment boundedness of linear stochastic delay differential equations

 1 School of Mathematics, Hefei University of Technology, Hefei 230009, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 3 Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing, 100084

Received  July 2013 Revised  March 2014 Published  September 2014

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework--for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay--of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a specific case of a type of $2$-dimensional equation that the stochastic terms are decoupled. For the $2$-dimensional equation, we obtain the characteristic function that is explicitly given by equation coefficients, and the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.
Citation: Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963
##### References:
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##### References:
 [1] J. A. D. Appleby, X. Mao and H. Wu, On the almost sure running maxima of solutions of affine stochastic functional differential equations, Siam J. Math. Anal., 42 (2010), 646-678. doi: 10.1137/080738404. [2] O. Arino, M. L. Hbid and E. Ait Dads, Delay Differential Equations and Applications, Proceedings of the NATO Advanced Study Institute held at the Cadi Ayyad University, Marrakech, 2002, (Edited by O. Arino, M. L. Hbid and E. Ait Dads), NATO Science Series II: Mathematics, Physics and Chemistry, 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7. [3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic, New York, 1963. [4] T. Caraballo, J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Advanced Nonlinear Studies, 10 (2010), 23-52. [5] J. Duan, K. Lu, and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. and Diff. Eqns., 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z. [6] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Press, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [7] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equations, J. Math. Kyoto Univ., 4 (1964), 1-75. [8] A. F. Ivanov, Y. I. Kazmerchuk and A. V. Swishchuk, Theory, stochastic stability and applications of stochastic delay differential equations: A survey of results, Differential Equations Dynamical Systems, 11 (2003), 55-115. [9] J. Lei and M. C. Mackey, Stochastic differential delay equation, Moment stability and its application to the hamatopoietic stem cell regulation system, SIAM J. Appl. Math., 67 (2007), 387-407. doi: 10.1137/060650234. [10] M. C. Mackey and I. G. Nechaeva, Noise and stability in differential delay equations, J. Dynam. and Diff. Eqns., 6 (1994), 395-426. doi: 10.1007/BF02218856. [11] M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations, Phs. Rev. E, 52 (1995), 3366-3376. doi: 10.1103/PhysRevE.52.3366. [12] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, UK, 1997. [13] X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215-227. doi: 10.1016/S0377-0427(02)00750-1. [14] X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis, 47 (2001), 4795-4806. doi: 10.1016/S0362-546X(01)00591-0. [15] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23(2005), 1045-1069. doi: 10.1080/07362990500118637. [16] X. Mao and M. J. Rassias, Almost sure asymptotic estimations for solutions of stochastic differential delay equations, Int. J. Appl. Math. Stat., 9 (2007), 95-109. [17] S.-E. A. Mohammed, Stochastic Functional Differential Equations, Res. Notes in Math. 99, Pitman, Boston, 1984. [18] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow, J. Funct. Anal., 205 (2003), 271-305. doi: 10.1016/j.jfa.2002.04.001. [19] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory II. The local stable manifold theorem, J. Funct. Anal., 206 (2004), 253-306. doi: 10.1016/j.jfa.2003.06.002. [20] Z. Wang, X. Li and J. Lei, Moment boundedness of linear stochastic differential equations, Stoch. Proc. Appl., 124 (2014), 586-612. doi: 10.1016/j.spa.2013.09.002.
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