# 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), 589-610. 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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##### 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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