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. ApplebyX. 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. BaoA. 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. BaoG. 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. HouJ. 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. JankovićJ. 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. MaoY. 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. ReißM. 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. TanW. 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. YuanJ. 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. ZhouJ. 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. ApplebyX. 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. BaoA. 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. BaoG. 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. HouJ. 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. JankovićJ. 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. MaoY. 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. ReißM. 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. TanW. 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. YuanJ. 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. ZhouJ. 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

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