American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator
  • DCDS-B Home
  • This Issue
  • Next Article
    Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise
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.  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

[1]

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
[2]

Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325
[3]

Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110
[4]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[5]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[6]

Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069
[7]

Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[8]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[9]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[10]

François Lalonde, Egor Shelukhin. Proof of the main conjecture on $g$-areas. Electronic Research Announcements, 2015, 22: 92-102. doi: 10.3934/era.2015.22.92
[11]

Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
[12]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517
[13]

Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[14]

Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867
[15]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001
[16]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615
[17]

Fadia Bekkal-Brikci, Giovanna Chiorino, Khalid Boushaba. G1/S transition and cell population dynamics. Networks & Heterogeneous Media, 2009, 4 (1) : 67-90. doi: 10.3934/nhm.2009.4.67
[18]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008
[19]

Minvydas Ragulskis, Zenonas Navickas. Hash function construction based on time average moiré. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 1007-1020. doi: 10.3934/dcdsb.2007.8.1007
[20]

Umberto Mosco. Impulsive motion on synchronized spatial temporal grids. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6069-6098. doi: 10.3934/dcds.2017261

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (65)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences