# American Institute of Mathematical Sciences

September  2015, 20(7): 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

## Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion

 1 Department of Mathematics, Anhui Normal University, Wuhu 241000, Anhui, China, China 2 Department of Mathematics, Anhui Normal University, Wuhu 241000, Anhui,, China

Received  August 2014 Revised  March 2015 Published  July 2015

In this paper, we establish the $p$-th moment exponential stability and quasi sure exponential stability of the solutions to impulsive stochastic differential equations driven by $G$-Brownian motion (IGSDEs in short) by means of $G$-Lyapunov function method. An example is presented to illustrate the efficiency of the obtained results.
Citation: 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
##### References:
 [1] X. Bai and Y. Lin, On the existence and uniqunenss of solutions to the stochastic differential equations driven by $G$-Brownian motion with integral lipschitz codfficients,, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 589. doi: 10.1007/s10255-014-0405-9. [2] Z. Chen, Strong laws of large number for capacities,, preprint, (). [3] L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion pathes,, Potential Anal., 34 (2011), 139. doi: 10.1007/s11118-010-9185-x. [4] W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion,, preprint, (). [5] W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion,, preprint, (). [6] F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion,, Stochastic Process. Appl., 119 (2009), 3356. doi: 10.1016/j.spa.2009.05.010. [7] M. Hu and S. Peng, On the representation theorem of $G$-expectations and paths of $G$-Brownian motion,, Acta Math. Appl. Sin. Engl. Ser., 25 (2009), 539. doi: 10.1007/s10255-008-8831-1. [8] L. Hu, Y. Ren and T. Xu, $p$-moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion,, Appl. Math. Comput., 230 (2014), 231. doi: 10.1016/j.amc.2013.12.111. [9] V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations,, World Scientific, (1989). doi: 10.1142/0906. [10] X. Li and S. Peng, Stopping times and related Itô's calcilus with $G$-Brownian motion,, Stochastic Process. Appl., 121 (2011), 1492. doi: 10.1016/j.spa.2011.03.009. [11] Y. Lin, Stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions,, Electronic. J. Probbab., 18 (2013). doi: 10.1214/EJP.v18-2566. [12] X. Liu, Impulsive stabilization of nonlinear systems,, IMA J. Math. Control Inform., 10 (1993), 11. doi: 10.1093/imamci/10.1.11. [13] B. Liu, Stability of solutions for stochastic impulsive systems via comparison approach,, IEEE Trans. Automat. Control, 53 (2008), 2128. doi: 10.1109/TAC.2008.930185. [14] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type,, in Stochastic Analysis and Applications, (2007), 541. doi: 10.1007/978-3-540-70847-6_25. [15] S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty,, preprint, (). [16] S. Peng, Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation,, Stochastic Process. Appl., 118 (2008), 2223. doi: 10.1016/j.spa.2007.10.015. [17] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty,, preprint, (). [18] S. Peng and B. Jia, Some criteria on $p$-th moment stability of impulsive stochastic functional differential equations,, Statist. Probab. Lett., 80 (2010), 1085. doi: 10.1016/j.spl.2010.03.002. [19] Y. Ren, Q. Bi and R. Sakthivel, Stochastic functional differential equations with infinite delay driven by $G$-Brownian motion,, Math. Methods Appl. Sci., 36 (2013), 1746. doi: 10.1002/mma.2720. [20] Y. Ren and L. Hu, A note on the stochastic differential equations driven by $G$-Brownian motion,, Statist. Probab. Lett., 81 (2011), 580. doi: 10.1016/j.spl.2011.01.010. [21] L. Shen and J. Sun, $p$-th moment exponential stability of stochastic differential equations with impulsive effect,, Sci. China Inf. Sci., 54 (2011), 1702. doi: 10.1007/s11432-011-4250-7. [22] S. Wu, D. Han and X. Meng, $p$-moment stability of stochastic differential equations with jumps,, Appl. Math. Comput., 152 (2004), 505. doi: 10.1016/S0096-3003(03)00573-3. [23] H. Wu and J. Sun, $p$-moment stability of stochastic differential equations with impulsive jump and Markovian switching,, Automatica J. IFAC, 42 (2006), 1753. doi: 10.1016/j.automatica.2006.05.009. [24] X. Wu, L. Yan, W. Zhang and L. Chen, Exponential stability of impulsive stochastic delay differential systems,, Discrete Dyn. Nat. Soc., (2012). doi: 10.1155/2012/296136. [25] D. Zhang and Z. Chen, Exponential stability for stochastic differential equations driven by $G$-Brownian motion,, Appl. Math. Lett., 25 (2012), 1906. doi: 10.1016/j.aml.2012.02.063. [26] B. Zhang, J. Xu and D. Kannan, Extension and application of Itô's formula under $G$-framework,, Stochastic Anal. Appl., 28 (2010), 322. doi: 10.1080/07362990903546595.

show all references

##### References:
 [1] X. Bai and Y. Lin, On the existence and uniqunenss of solutions to the stochastic differential equations driven by $G$-Brownian motion with integral lipschitz codfficients,, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 589. doi: 10.1007/s10255-014-0405-9. [2] Z. Chen, Strong laws of large number for capacities,, preprint, (). [3] L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion pathes,, Potential Anal., 34 (2011), 139. doi: 10.1007/s11118-010-9185-x. [4] W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion,, preprint, (). [5] W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion,, preprint, (). [6] F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion,, Stochastic Process. Appl., 119 (2009), 3356. doi: 10.1016/j.spa.2009.05.010. [7] M. Hu and S. Peng, On the representation theorem of $G$-expectations and paths of $G$-Brownian motion,, Acta Math. Appl. Sin. Engl. Ser., 25 (2009), 539. doi: 10.1007/s10255-008-8831-1. [8] L. Hu, Y. Ren and T. Xu, $p$-moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion,, Appl. Math. Comput., 230 (2014), 231. doi: 10.1016/j.amc.2013.12.111. [9] V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations,, World Scientific, (1989). doi: 10.1142/0906. [10] X. Li and S. Peng, Stopping times and related Itô's calcilus with $G$-Brownian motion,, Stochastic Process. Appl., 121 (2011), 1492. doi: 10.1016/j.spa.2011.03.009. [11] Y. Lin, Stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions,, Electronic. J. Probbab., 18 (2013). doi: 10.1214/EJP.v18-2566. [12] X. Liu, Impulsive stabilization of nonlinear systems,, IMA J. Math. Control Inform., 10 (1993), 11. doi: 10.1093/imamci/10.1.11. [13] B. Liu, Stability of solutions for stochastic impulsive systems via comparison approach,, IEEE Trans. Automat. Control, 53 (2008), 2128. doi: 10.1109/TAC.2008.930185. [14] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type,, in Stochastic Analysis and Applications, (2007), 541. doi: 10.1007/978-3-540-70847-6_25. [15] S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty,, preprint, (). [16] S. Peng, Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation,, Stochastic Process. Appl., 118 (2008), 2223. doi: 10.1016/j.spa.2007.10.015. [17] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty,, preprint, (). [18] S. Peng and B. Jia, Some criteria on $p$-th moment stability of impulsive stochastic functional differential equations,, Statist. Probab. Lett., 80 (2010), 1085. doi: 10.1016/j.spl.2010.03.002. [19] Y. Ren, Q. Bi and R. Sakthivel, Stochastic functional differential equations with infinite delay driven by $G$-Brownian motion,, Math. Methods Appl. Sci., 36 (2013), 1746. doi: 10.1002/mma.2720. [20] Y. Ren and L. Hu, A note on the stochastic differential equations driven by $G$-Brownian motion,, Statist. Probab. Lett., 81 (2011), 580. doi: 10.1016/j.spl.2011.01.010. [21] L. Shen and J. Sun, $p$-th moment exponential stability of stochastic differential equations with impulsive effect,, Sci. China Inf. Sci., 54 (2011), 1702. doi: 10.1007/s11432-011-4250-7. [22] S. Wu, D. Han and X. Meng, $p$-moment stability of stochastic differential equations with jumps,, Appl. Math. Comput., 152 (2004), 505. doi: 10.1016/S0096-3003(03)00573-3. [23] H. Wu and J. Sun, $p$-moment stability of stochastic differential equations with impulsive jump and Markovian switching,, Automatica J. IFAC, 42 (2006), 1753. doi: 10.1016/j.automatica.2006.05.009. [24] X. Wu, L. Yan, W. Zhang and L. Chen, Exponential stability of impulsive stochastic delay differential systems,, Discrete Dyn. Nat. Soc., (2012). doi: 10.1155/2012/296136. [25] D. Zhang and Z. Chen, Exponential stability for stochastic differential equations driven by $G$-Brownian motion,, Appl. Math. Lett., 25 (2012), 1906. doi: 10.1016/j.aml.2012.02.063. [26] B. Zhang, J. Xu and D. Kannan, Extension and application of Itô's formula under $G$-framework,, Stochastic Anal. Appl., 28 (2010), 322. doi: 10.1080/07362990903546595.
 [1] Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019110 [2] 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 [3] Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 [4] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 [5] 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 [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] 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 [8] 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 [9] 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 [10] 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 [11] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [12] 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 [13] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [14] 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 [15] 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 [16] Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683 [17] Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801 [18] Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219 [19] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [20] 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

2018 Impact Factor: 1.008