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

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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

Yong Ren ^, , Wensheng Yin and Dongjin Zhu

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.

Keywords: G-Brownian motion, G-Lyapunov function, delayed impulses, average impulsive interval, exponential stability.

Mathematics Subject Classification: Primary: 93E15, 34K50, 60H10.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. Peng, G-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834v1.
[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.
[15]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv: 1002.4546v1.
[16]	S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. Peng, G-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834v1.
[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.
[15]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv: 1002.4546v1.
[16]	S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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