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Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise
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 |
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.
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. |
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. |
[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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