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October  2018, 23(8): 3347-3360. doi: 10.3934/dcdsb.2018248

## Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach

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

* Corresponding author: Yong Ren

Received  January 2018 Published  August 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 11371029 and 11501009.

In this article, we discuss a class of impulsive stochastic function differential equations driven by $G$-Brownian motion with delayed impulsive effects ($G$-DISFDEs, in short). Some sufficient conditions for $p$-th moment exponential stability of $G$-DISFDEs are derived by means of $G$-Lyapunov function method, average impulsive interval approach and Razumikhin-type conditions. An example is provided to show the effectiveness of the theoretical results.

Citation: 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
##### References:
 [1] P. Cheng, F. Deng and F. Yao, Exponential stability analysis of impulsive stochastic functional differential systems with delayed impules, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2104-2114.  doi: 10.1016/j.cnsns.2013.10.008.  Google Scholar [2] 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 [3] 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 [4] F. Hu, Z. Chen and P. Wu, A general strong law of large numbers for non-additive probabilities and its applications, Statistics, 50 (2016), 733-749.  doi: 10.1080/02331888.2016.1143473.  Google Scholar [5] F. Hu, Z. Chen and D. Zhang, How big are the increments of $G$-Brownian motion?, Sci. China Math., 57 (2014), 1687-1700.  doi: 10.1007/s11425-014-4816-0.  Google Scholar [6] F. Hu and Z. Chen, General laws of large numbers under sublinear expectations, Comm. Statist. Theory Methods, 45 (2016), 4215-4229.  doi: 10.1080/03610926.2014.917677.  Google Scholar [7] L. Hu and Y. Ren, Implusive stochastic differential equations driven by G-Brownian motion, In Brownian Motion: Elements, Dynamics and Applications, editors: Mark A. McKibben and Micah Webster, Nova Science Publishers, Inc, New York, 2015, Capter 13,231–242. Google Scholar [8] 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 [9] D. Li, P. Cheng and S. Shu, Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach, Math. Methods Appl. Sci., 40 (2017), 4197-4210.  doi: 10.1002/mma.4297.  Google Scholar [10] X. Li and S. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stochastic Process. Appl., 121 (2011), 1492-1508.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar [11] X. Li, X. Lin and Y. Lin, Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion, J. Math. Anal. Appl., 439 (2016), 235-255.  doi: 10.1016/j.jmaa.2016.02.042.  Google Scholar [12] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., Springer, Berlin, 2 (2007), 541–567. doi: 10.1007/978-3-540-70847-6_25.  Google Scholar [13] S. Peng, G-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834v1. Google Scholar [14] 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 [15] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv: 1002.4546v1. Google Scholar [16] S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254.   Google Scholar [17] Y. Ren, Q. Bi and R. Sakthivel, Stochastic functional differential equations with infinite delay driven by $G$-Brownian motion, Math. Method. Appl. Sci., 36 (2013), 1746-1759.  doi: 10.1002/mma.2720.  Google Scholar [18] 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 [19] Y. Ren, X. Jia and L. Hu, Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, iscrete Conti. Dyn. Syst. Ser-B., 20 (2015), 2157-2169.  doi: 10.3934/dcdsb.2015.20.2157.  Google Scholar [20] Y. Ren, X. Jia and R. Sakthivel, The $p$-th moment stability of solution to impulsive stochastic differential equations driven by $G$-Brownian motion, Appl. Anal., 96 (2017), 988-1003.  doi: 10.1080/00036811.2016.1169529.  Google Scholar [21] Y. Ren, J. Wang and L. Hu, Multi-valued stochastic differential equations driven by $G$-Brownian motion and related stochastic control problems, Internat. J. Control, 90 (2017), 1132-1154.  doi: 10.1080/00207179.2016.1204560.  Google Scholar [22] F. Yao, J. Cao, L. Qiu and P. Cheng, Exponential stability analysis for stochastic delayed differential systems with impulsive effects: average impulsive interval approach, Asian J. Control, 19 (2017), 74-86.  doi: 10.1002/asjc.1320.  Google Scholar [23] W. Yin and Y. Ren, Asymptotical boundedness and stability for stochastic differential equations with delay driven by $G$-Brownian motion, Appl. Math. Lett., 74 (2017), 121-126.  doi: 10.1016/j.aml.2017.06.001.  Google Scholar [24] 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 [25] D. Zhang and P. He, Functional solution about stochastic differential equations driven by $G$-Brownian motion, Discrete Conti. Dyn. Syst. Ser-B., 20 (2015), 281-293.  doi: 10.3934/dcdsb.2015.20.281.  Google Scholar

show all references

##### References:
 [1] P. Cheng, F. Deng and F. Yao, Exponential stability analysis of impulsive stochastic functional differential systems with delayed impules, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2104-2114.  doi: 10.1016/j.cnsns.2013.10.008.  Google Scholar [2] 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 [3] 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 [4] F. Hu, Z. Chen and P. Wu, A general strong law of large numbers for non-additive probabilities and its applications, Statistics, 50 (2016), 733-749.  doi: 10.1080/02331888.2016.1143473.  Google Scholar [5] F. Hu, Z. Chen and D. Zhang, How big are the increments of $G$-Brownian motion?, Sci. China Math., 57 (2014), 1687-1700.  doi: 10.1007/s11425-014-4816-0.  Google Scholar [6] F. Hu and Z. Chen, General laws of large numbers under sublinear expectations, Comm. Statist. Theory Methods, 45 (2016), 4215-4229.  doi: 10.1080/03610926.2014.917677.  Google Scholar [7] L. Hu and Y. Ren, Implusive stochastic differential equations driven by G-Brownian motion, In Brownian Motion: Elements, Dynamics and Applications, editors: Mark A. McKibben and Micah Webster, Nova Science Publishers, Inc, New York, 2015, Capter 13,231–242. Google Scholar [8] 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 [9] D. Li, P. Cheng and S. Shu, Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach, Math. Methods Appl. Sci., 40 (2017), 4197-4210.  doi: 10.1002/mma.4297.  Google Scholar [10] X. Li and S. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stochastic Process. Appl., 121 (2011), 1492-1508.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar [11] X. Li, X. Lin and Y. Lin, Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion, J. Math. Anal. Appl., 439 (2016), 235-255.  doi: 10.1016/j.jmaa.2016.02.042.  Google Scholar [12] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., Springer, Berlin, 2 (2007), 541–567. doi: 10.1007/978-3-540-70847-6_25.  Google Scholar [13] S. Peng, G-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834v1. Google Scholar [14] 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 [15] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv: 1002.4546v1. Google Scholar [16] S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254.   Google Scholar [17] Y. Ren, Q. Bi and R. Sakthivel, Stochastic functional differential equations with infinite delay driven by $G$-Brownian motion, Math. Method. Appl. Sci., 36 (2013), 1746-1759.  doi: 10.1002/mma.2720.  Google Scholar [18] 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 [19] Y. Ren, X. Jia and L. Hu, Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, iscrete Conti. Dyn. Syst. Ser-B., 20 (2015), 2157-2169.  doi: 10.3934/dcdsb.2015.20.2157.  Google Scholar [20] Y. Ren, X. Jia and R. Sakthivel, The $p$-th moment stability of solution to impulsive stochastic differential equations driven by $G$-Brownian motion, Appl. Anal., 96 (2017), 988-1003.  doi: 10.1080/00036811.2016.1169529.  Google Scholar [21] Y. Ren, J. Wang and L. Hu, Multi-valued stochastic differential equations driven by $G$-Brownian motion and related stochastic control problems, Internat. J. Control, 90 (2017), 1132-1154.  doi: 10.1080/00207179.2016.1204560.  Google Scholar [22] F. Yao, J. Cao, L. Qiu and P. Cheng, Exponential stability analysis for stochastic delayed differential systems with impulsive effects: average impulsive interval approach, Asian J. Control, 19 (2017), 74-86.  doi: 10.1002/asjc.1320.  Google Scholar [23] W. Yin and Y. Ren, Asymptotical boundedness and stability for stochastic differential equations with delay driven by $G$-Brownian motion, Appl. Math. Lett., 74 (2017), 121-126.  doi: 10.1016/j.aml.2017.06.001.  Google Scholar [24] 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 [25] D. Zhang and P. He, Functional solution about stochastic differential equations driven by $G$-Brownian motion, Discrete Conti. Dyn. Syst. Ser-B., 20 (2015), 281-293.  doi: 10.3934/dcdsb.2015.20.281.  Google Scholar
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