American Institute of Mathematical Sciences

Advanced
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

Yong Ren 1, , Xuejuan Jia 1, and  Lanying Hu 2,

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.
Keywords: Stochastic differential equation, $G$-Lyapunov function method., $G$-Brownian motion, quasi sure exponential stability, $p$-th moment exponential stability.
Mathematics Subject Classification: Primary: 34A37, 60H10; Secondary: 34D1.
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.  Google Scholar

[2]

Z. Chen, Strong laws of large number for capacities,, preprint, ().   Google Scholar

[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.  Google Scholar

[4]

W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion,, preprint, ().   Google Scholar

[5]

W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations,, World Scientific, (1989).  doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. Liu, Impulsive stabilization of nonlinear systems,, IMA J. Math. Control Inform., 10 (1993), 11.  doi: 10.1093/imamci/10.1.11.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty,, preprint, ().   Google Scholar

[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.  Google Scholar

[17]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

Z. Chen, Strong laws of large number for capacities,, preprint, ().   Google Scholar

[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.  Google Scholar

[4]

W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion,, preprint, ().   Google Scholar

[5]

W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations,, World Scientific, (1989).  doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. Liu, Impulsive stabilization of nonlinear systems,, IMA J. Math. Control Inform., 10 (1993), 11.  doi: 10.1093/imamci/10.1.11.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty,, preprint, ().   Google Scholar

[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.  Google Scholar

[17]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. 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]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[7]

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
[8]

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
[9]

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
[10]

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
[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]

Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
[13]

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
[14]

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
[15]

Wensheng Yin, Jinde Cao. Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020109
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences