# 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:
 [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.

show all references

##### 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.
 [1] Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332 [2] Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042 [3] William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020 [4] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [5] Ekaterina Gromova, Ekaterina Marova, Dmitry Gromov. A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games. Journal of Dynamics and Games, 2020, 7 (2) : 105-122. doi: 10.3934/jdg.2020007 [6] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [7] Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044 [8] Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems and Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25 [9] Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349 [10] Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems and Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317 [11] Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147 [12] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432 [13] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [14] Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1 [15] Yufeng Zhang, Wen-Xiu Ma, Jin-Yun Yang. A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2941-2948. doi: 10.3934/dcdss.2020167 [16] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [17] Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049 [18] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501 [19] Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301 [20] Elena Braverman, Karel Hasik, Anatoli F. Ivanov, Sergei I. Trofimchuk. A cyclic system with delay and its characteristic equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 1-29. doi: 10.3934/dcdss.2020001

2020 Impact Factor: 1.327