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

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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

Abstract
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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 and 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.
[2]	Z. Chen, Strong laws of large number for capacities, preprint, arXiv:1006.0749.
[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.
[4]	W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion, preprint, arXiv:1309.0209.
[5]	W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion, preprint, arXiv:1311.7311.
[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.
[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.
[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.
[9]	V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 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), 1492-1508. 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), 23pp. doi: 10.1214/EJP.v18-2566.
[12]	X. Liu, Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10 (1993), 11-19. 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), 2128-2133. 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, Abel Symp., 2, Springer, Berlin, 2007, 541-567. doi: 10.1007/978-3-540-70847-6_25.
[15]	S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv:0711.2834.
[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.
[17]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv:1002.4546.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[2]	Z. Chen, Strong laws of large number for capacities, preprint, arXiv:1006.0749.
[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.
[4]	W. Fei and C. Fei, Optimal stochastic control and optimal consumption and portfolio with $G$-Brownian motion, preprint, arXiv:1309.0209.
[5]	W. Fei and C. Fei, Exponential stability for stochastic differential equations disturbed by $G$-Brownian motion, preprint, arXiv:1311.7311.
[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.
[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.
[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.
[9]	V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 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), 1492-1508. 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), 23pp. doi: 10.1214/EJP.v18-2566.
[12]	X. Liu, Impulsive stabilization of nonlinear systems, IMA J. Math. Control Inform., 10 (1993), 11-19. 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), 2128-2133. 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, Abel Symp., 2, Springer, Berlin, 2007, 541-567. doi: 10.1007/978-3-540-70847-6_25.
[15]	S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv:0711.2834.
[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.
[17]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv:1002.4546.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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