# American Institute of Mathematical Sciences

September  2017, 22(7): 2923-2938. doi: 10.3934/dcdsb.2017157

## Exponential stability of solutions for retarded stochastic differential equations without dissipativity

 1 College of Traffic Engineering, Hunan University of Technology, Zhuzhou, Hunan 412007, China 2 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China 3 Department of Mathematics, Swansea University, Singleton Park, SA2, 8PP, UK

* Corresponding author: Min Zhu

Received  July 2016 Revised  April 2017 Published  May 2017

This work focuses on a class of retarded stochastic differential equations that need not satisfy dissipative conditions. The principle technique of our investigation is to use variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. By using variation-of-constants formula and estimating the diffusion coefficients we give sufficient conditions for $p$-th moment exponential stability, almost sure exponential stability and convergence of solutions from different initial value. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.

Citation: Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
##### 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 Journal on Mathematical Analysis, 42 (2010), 646-678. doi: 10.1137/080738404. Google Scholar [2] J. A. D. Appleby, H. Wu and X. Mao, On the almost sure running maxima of solutions of affine neutral stochastic functional differential equations preprint, arXiv: 1310.2349 (2013).Google Scholar [3] J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society,, 465 (2009), 2111-2134. doi: 10.1098/rspa.2008.0486. Google Scholar [4] J. Bao and C. Yuan, Stationary distributions for retarded stochastic differential equations without dissipativity, Stochastic, (2016), 1-20. Google Scholar [5] J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Analysis: Theory, Methods and Applications, 98 (2014), 66-82. doi: 10.1016/j.na.2013.12.001. Google Scholar [6] J. Bao, G. Yin and C. Yuan, Asymptotic Analysis for Functional Stochastic Equations Spinger, 2016. doi: 10.1007/978-3-319-46979-9. Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [8] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [9] Z. Hou, J. Bao and C. Yuan, Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps, Journal of Mathematical Analysis and Applications, 366 (2010), 44-54. doi: 10.1016/j.jmaa.2010.01.019. Google Scholar [10] S. Janković, J. Randjelovic and M. Jovanović, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 355 (2009), 811-820. doi: 10.1016/j.jmaa.2009.02.011. Google Scholar [11] K. Liu, The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Applied Mathematics and Optimization, 60 (2009), 1-38. doi: 10.1007/s00245-009-9065-1. Google Scholar [12] K. Liu and Y. Shi, Razumikhin-type theorems of infinite dimensional stochastic functional differential equations, in IFIP Conference on System Modeling and Optimization, Springer US, 202 (2006), 237-247. doi: 10.1007/0-387-33882-9_22. Google Scholar [13] X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems & Control Letters, 26 (1995), 245-251. doi: 10.1016/0167-6911(95)00018-5. Google Scholar [14] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Processes and their Applications, 65 (1996), 233-250. doi: 10.1016/S0304-4149(96)00109-3. Google Scholar [15] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM Journal on Mathematical Analysis, 28 (1997), 389-401. doi: 10.1137/S0036141095290835. Google Scholar [16] X. Mao, Y. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications, 118 (2008), 1385-1406. doi: 10.1016/j.spa.2007.09.005. Google Scholar [17] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 268 (2002), 125-142. doi: 10.1006/jmaa.2001.7803. Google Scholar [18] X. Mao, Stochastic Differential Equationa and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar [19] S. -E. A. Mohammed, Stochastic Functional Differential Equations Pitman, Boston, 1984. Google Scholar [20] D. Nguyen, Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 1-7. doi: 10.1016/j.cnsns.2013.06.004. Google Scholar [21] M. Reiß, M. Riedle and O. Gaans, Delay differential equations driven by Lévy processes: stationarity and Feller properties, Stochastic Processes and their Applications, 116 (2006), 1409-1432. doi: 10.1016/j.spa.2006.03.002. Google Scholar [22] L. Tan, W. Jin and Y. Suo, Stability in distribution of neutral stochastic functional differential equations, Stochastics and Probability Letters, 107 (2015), 27-36. doi: 10.1016/j.spl.2015.07.033. Google Scholar [23] F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete and Continuous Dynamical System Series-B, 18 (2013), 1715-1734. doi: 10.3934/dcdsb.2013.18.1715. Google Scholar [24] C. Yuan, J. Zou and X. Mao, tability in distribution of stochastic differential delay equations with Markovian switching, Systems and Control Letters, 50 (2003), 195-207. doi: 10.1016/S0167-6911(03)00154-3. Google Scholar [25] W. Zhou, J. Yang and X. Yang, p-th moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Applied Mathematical Modelling, 39 (2015), 5650-5658. doi: 10.1016/j.apm.2015.01.025. Google Scholar [26] Q. Zhu, Asymptotic stability in the p-th moment for stochastic differential equations with Lévy noise, Journal of Mathematical Analysis and Applications, 416 (2014), 126-142. doi: 10.1016/j.jmaa.2014.02.016. Google Scholar [27] X. Zong and F. Wu, Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations, Applied Mathematical Modelling, 40 (2016), 19-30. doi: 10.1016/j.apm.2015.05.001. Google Scholar

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 Journal on Mathematical Analysis, 42 (2010), 646-678. doi: 10.1137/080738404. Google Scholar [2] J. A. D. Appleby, H. Wu and X. Mao, On the almost sure running maxima of solutions of affine neutral stochastic functional differential equations preprint, arXiv: 1310.2349 (2013).Google Scholar [3] J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society,, 465 (2009), 2111-2134. doi: 10.1098/rspa.2008.0486. Google Scholar [4] J. Bao and C. Yuan, Stationary distributions for retarded stochastic differential equations without dissipativity, Stochastic, (2016), 1-20. Google Scholar [5] J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Analysis: Theory, Methods and Applications, 98 (2014), 66-82. doi: 10.1016/j.na.2013.12.001. Google Scholar [6] J. Bao, G. Yin and C. Yuan, Asymptotic Analysis for Functional Stochastic Equations Spinger, 2016. doi: 10.1007/978-3-319-46979-9. Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [8] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [9] Z. Hou, J. Bao and C. Yuan, Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps, Journal of Mathematical Analysis and Applications, 366 (2010), 44-54. doi: 10.1016/j.jmaa.2010.01.019. Google Scholar [10] S. Janković, J. Randjelovic and M. Jovanović, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 355 (2009), 811-820. doi: 10.1016/j.jmaa.2009.02.011. Google Scholar [11] K. Liu, The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Applied Mathematics and Optimization, 60 (2009), 1-38. doi: 10.1007/s00245-009-9065-1. Google Scholar [12] K. Liu and Y. Shi, Razumikhin-type theorems of infinite dimensional stochastic functional differential equations, in IFIP Conference on System Modeling and Optimization, Springer US, 202 (2006), 237-247. doi: 10.1007/0-387-33882-9_22. Google Scholar [13] X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems & Control Letters, 26 (1995), 245-251. doi: 10.1016/0167-6911(95)00018-5. Google Scholar [14] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Processes and their Applications, 65 (1996), 233-250. doi: 10.1016/S0304-4149(96)00109-3. Google Scholar [15] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM Journal on Mathematical Analysis, 28 (1997), 389-401. doi: 10.1137/S0036141095290835. Google Scholar [16] X. Mao, Y. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications, 118 (2008), 1385-1406. doi: 10.1016/j.spa.2007.09.005. Google Scholar [17] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 268 (2002), 125-142. doi: 10.1006/jmaa.2001.7803. Google Scholar [18] X. Mao, Stochastic Differential Equationa and Applications 2nd edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar [19] S. -E. A. Mohammed, Stochastic Functional Differential Equations Pitman, Boston, 1984. Google Scholar [20] D. Nguyen, Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 1-7. doi: 10.1016/j.cnsns.2013.06.004. Google Scholar [21] M. Reiß, M. Riedle and O. Gaans, Delay differential equations driven by Lévy processes: stationarity and Feller properties, Stochastic Processes and their Applications, 116 (2006), 1409-1432. doi: 10.1016/j.spa.2006.03.002. Google Scholar [22] L. Tan, W. Jin and Y. Suo, Stability in distribution of neutral stochastic functional differential equations, Stochastics and Probability Letters, 107 (2015), 27-36. doi: 10.1016/j.spl.2015.07.033. Google Scholar [23] F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete and Continuous Dynamical System Series-B, 18 (2013), 1715-1734. doi: 10.3934/dcdsb.2013.18.1715. Google Scholar [24] C. Yuan, J. Zou and X. Mao, tability in distribution of stochastic differential delay equations with Markovian switching, Systems and Control Letters, 50 (2003), 195-207. doi: 10.1016/S0167-6911(03)00154-3. Google Scholar [25] W. Zhou, J. Yang and X. Yang, p-th moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Applied Mathematical Modelling, 39 (2015), 5650-5658. doi: 10.1016/j.apm.2015.01.025. Google Scholar [26] Q. Zhu, Asymptotic stability in the p-th moment for stochastic differential equations with Lévy noise, Journal of Mathematical Analysis and Applications, 416 (2014), 126-142. doi: 10.1016/j.jmaa.2014.02.016. Google Scholar [27] X. Zong and F. Wu, Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations, Applied Mathematical Modelling, 40 (2016), 19-30. doi: 10.1016/j.apm.2015.05.001. Google Scholar
 [1] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [2] Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571 [3] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [4] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [5] Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219 [6] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [7] Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721 [8] Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 [9] Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517 [10] Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555 [11] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [12] Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093 [13] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [14] Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 [15] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [16] Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657 [17] Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003 [18] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [19] Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280 [20] John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (19)
• HTML views (25)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]