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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), 589-610. doi: 10.1007/s10255-014-0405-9.  Google Scholar

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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), 139-161. doi: 10.1007/s11118-010-9185-x.  Google Scholar

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

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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), 3356-3382. 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) 539-546. 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), 231-237. 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, Singapore, 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), 1492-1508. 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), 23pp. doi: 10.1214/EJP.v18-2566.  Google Scholar

[12]

X. Liu, Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10 (1993), 11-19. 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), 2128-2133. 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, Abel Symp., 2, Springer, Berlin, 2007, 541-567. 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), 2223-2253. 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), 1085-1092. 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), 1746-1759. doi: 10.1002/mma.2720.  Google Scholar

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Y. Ren and L. Hu, A note on the stochastic differential equations driven by $G$-Brownian motion, Statist. Probab. Lett., 81 (2011), 580-585. 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), 1702-1711. 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), 505-519. 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), 1753-1759. 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), Art. ID 296136, 15pp. 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), 1906-1910. 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), 322-349. 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), 589-610. 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), 139-161. 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), 3356-3382. 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) 539-546. 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), 231-237. 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, Singapore, 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), 1492-1508. 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), 23pp. doi: 10.1214/EJP.v18-2566.  Google Scholar

[12]

X. Liu, Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10 (1993), 11-19. 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), 2128-2133. 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, Abel Symp., 2, Springer, Berlin, 2007, 541-567. 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), 2223-2253. 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), 1085-1092. 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), 1746-1759. 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), 580-585. 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), 1702-1711. 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), 505-519. 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), 1753-1759. 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), Art. ID 296136, 15pp. 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), 1906-1910. 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), 322-349. doi: 10.1080/07362990903546595.  Google Scholar

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