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April  2020, 25(4): 1583-1606. doi: 10.3934/dcdsb.2019241

G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients

1. 

Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt

2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Mahmoud Abouagwa

Received  January 2019 Revised  July 2019 Published  April 2020 Early access  November 2019

Fund Project: The research of Ji Li is supported by the National Natural Science Foundation of China No. 11771161.

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by $ G $-Brownian motion (VNSDDEGs in short) under global Carathéodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in $ L^p $-sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.

Citation: Mahmoud Abouagwa, Ji Li. G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1583-1606. doi: 10.3934/dcdsb.2019241
References:
[1]

M. Abouagwa and J. Li, Approximation properties of solutions to Itô-Doob stochastic fractional differential equations with non-Lipschitz coefficents, Stoch. Dyn., 19 (2019), 1950029, 21 pp. doi: 10.1142/S0219493719500291.

[2]

M. Abouagwa and J. Li, Stochastic fractional differential equations driven by Lévy noise under Carathéodory conditions, J. Math. Phys., 60 (2019), 022701, 16 pp. doi: 10.1063/1.5063514.

[3]

M. AbouagwaJ. C. Liu and J. Li, Carathéodory approximations and stability of solutions to non-Lipschitz stochastic fractional differential equations of Itô-Doob type, Appl. Math. Comput., 329 (2018), 143-153.  doi: 10.1016/j.amc.2018.02.005.

[4]

X.-P Bai and Y.-P. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by $G$-Brownian motion with integral-Lipschitz coefficients, Acta Math. Appl. Sin. Eng. Ser., 30 (2014), 589-610.  doi: 10.1007/s10255-014-0405-9.

[5]

H. B. Bao and J. D. Cao, Existence and uniquenes of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732-1743.  doi: 10.1016/j.amc.2009.07.025.

[6]

L. DenisM. S. Hu and S. G. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Anal., 34 (2011), 139-161.  doi: 10.1007/s11118-010-9185-x.

[7]

F. Faizullah, M. Bux, M. A. Rana and G. ur Rahman, Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of $G$-Brownian motion, Adv. Differ. Equ., 2017 (2017), Paper No. 350, 14 pp. doi: 10.1186/s13662-017-1400-2.

[8]

F. Q. Gao, Pathwise propertise and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.

[9]

R. Guo and B. Pei, Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson point processes, Stoch. Anal. Appl., 36 (2018), 751-766.  doi: 10.1080/07362994.2018.1461567.

[10]

X. Y. He, S. Han and J. Tao, Averaging principle for SDEs of neutral type driven by $G$-Brownian motion, Stoch. Dyn., 19 (2019), 1950004, 22 pp. doi: 10.1142/S0219493719500047.

[11]

M.-S. Hu and S.-G. Peng, On representation theorem of $G$-expectations and paths of $G$-Brownian motion, Acta Math. Appl. Sin., Eng. Ser., 25 (2009), 539-546.  doi: 10.1007/s10255-008-8831-1.

[12]

L. Y. HuY. Ren and T. B. 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.

[13]

G. J. Li and Q. G. Yang, Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by $G$-Brownian motion, Comput. Appl. Math., 37 (2018), 4301-4320.  doi: 10.1007/s40314-018-0581-y.

[14]

Q. Lin, Local time and Tanaka formula for the $G$-Brownian motion, J. Math. Anal. Appl., 398 (2013), 315-334.  doi: 10.1016/j.jmaa.2012.09.001.

[15]

Q. Lin, Some properties of stochastic differential equations driven by $G$-Brownian motion, Acta Math. Appl. Sin. Eng. Ser., 29 (2013), 923-942.  doi: 10.1007/s10114-013-0701-y.

[16]

X. R. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optimiz., 12 (1991), 253-533.  doi: 10.1080/01630569108816448.

[17]

X. R. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Ⅱ, Numer. Funct. Anal. Optimiz., 15 (1994), 65-76.  doi: 10.1080/01630569408816550.

[18]

X. R. Mao, Approximate solutions for stochastic differntial equations with pathwise uniqueness, Stoch. Anal. Appl., 12 (1994), 355-367.  doi: 10.1080/07362999408809356.

[19]

X. R. Mao, Stochastic Differential Equations and Applications, 2nd edition, Ellis Horwood, Chichester, 2007.

[20]

W. Mao and X. R. Mao, An averaging principle for neutral stochastic differential equations driven by Poisson random measure, Adv. Differ. Equ., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0802-x.

[21]

S. Peng, $G$-Expectation, $G$-Brownian motion and related stochastic calculus of Itô type, in Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541–567.

[22]

S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834.

[23]

Y. RenQ. 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.

[24]

Y. Ren and L. Y. Hu, A note on the stochastic differential equations driven by $G$-Brownian motion, Stat. Prob. Lett., 81 (2011), 580-585.  doi: 10.1016/j.spl.2011.01.010.

[25]

Y. RenX. J. Jia and L. Y. Hu, Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2157-2169.  doi: 10.3934/dcdsb.2015.20.2157.

[26]

Y. RenS. P. Lu and N. M. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinte delay, J. Comput. Appl. Math., 220 (2008), 364-372.  doi: 10.1016/j.cam.2007.08.022.

[27]

Y. Ren and N. M. Xia, A note on the Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinte delay, Appl. Math. Comput., 214 (2009), 457-461.  doi: 10.1016/j.amc.2009.04.013.

[28]

Y. Ren and N. M. Xia, Existence, uniqueness and stability of solutions to neutral stochastic functional differential equations with infinte delay, Appl. Math. Comput., 210 (2009), 72-79.  doi: 10.1016/j.amc.2008.11.009.

[29]

Y. S. Song, Uniqueness of the representation for $G$-martingales with finite variation, Electron. J. Prob., 17 (2012), 15 pp. doi: 10.1214/EJP.v17-1890.

[30]

L. Tan and D. Y. Lei, The averaging method for stochastic differential delay equations under non-Lipschitz conditions, Adv. Differ. Equ., 38 (2013), 12 pp. doi: 10.1186/1687-1847-2013-38.

[31]

F. Y. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331 (2007), 516-531.  doi: 10.1016/j.jmaa.2006.09.020.

[32]

J. Xu and B. Zhang, Martingale characterization of $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 232-248.  doi: 10.1016/j.spa.2008.02.001.

[33]

Y. XuJ. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.

[34]

Y. XuB. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2257-2267.  doi: 10.3934/dcdsb.2015.20.2257.

[35]

Y. Xu, B. Pei and Y. G. Li, An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal., 2014 (2014), Art. ID 479195, 10 pp. doi: 10.1155/2014/479195.

[36]

Y. XuB. Pei and Y. G. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38 (2015), 2120-2131.  doi: 10.1002/mma.3208.

[37]

Y. Xu, B. Pei and J.-L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dyn., 17 (2017), 1750013, 16 pp. doi: 10.1142/S0219493717500137.

[38]

B. ZhangJ. Xu and D. Kannan, Extension and application of Itô's formula under $G$-framework, Stoch. Anal. Appl., 28 (2010), 322-349.  doi: 10.1080/07362990903546595.

[39]

D. F. Zhang and Z. J. Chen, Exponential stability for stochastic differential equation driven by $G$-Brownian motion, Appl. Math. Lett., 25 (2012), 1906-1910.  doi: 10.1016/j.aml.2012.02.063.

[40]

M. Zhang and G. F. Zong, Almost periodic solutions for stochastic differential equations driven by $G$-Brownian motion, Comm. Stat. Theor. Meth., 44 (2015), 2371-2384.  doi: 10.1080/03610926.2013.863935.

show all references

References:
[1]

M. Abouagwa and J. Li, Approximation properties of solutions to Itô-Doob stochastic fractional differential equations with non-Lipschitz coefficents, Stoch. Dyn., 19 (2019), 1950029, 21 pp. doi: 10.1142/S0219493719500291.

[2]

M. Abouagwa and J. Li, Stochastic fractional differential equations driven by Lévy noise under Carathéodory conditions, J. Math. Phys., 60 (2019), 022701, 16 pp. doi: 10.1063/1.5063514.

[3]

M. AbouagwaJ. C. Liu and J. Li, Carathéodory approximations and stability of solutions to non-Lipschitz stochastic fractional differential equations of Itô-Doob type, Appl. Math. Comput., 329 (2018), 143-153.  doi: 10.1016/j.amc.2018.02.005.

[4]

X.-P Bai and Y.-P. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by $G$-Brownian motion with integral-Lipschitz coefficients, Acta Math. Appl. Sin. Eng. Ser., 30 (2014), 589-610.  doi: 10.1007/s10255-014-0405-9.

[5]

H. B. Bao and J. D. Cao, Existence and uniquenes of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732-1743.  doi: 10.1016/j.amc.2009.07.025.

[6]

L. DenisM. S. Hu and S. G. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Anal., 34 (2011), 139-161.  doi: 10.1007/s11118-010-9185-x.

[7]

F. Faizullah, M. Bux, M. A. Rana and G. ur Rahman, Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of $G$-Brownian motion, Adv. Differ. Equ., 2017 (2017), Paper No. 350, 14 pp. doi: 10.1186/s13662-017-1400-2.

[8]

F. Q. Gao, Pathwise propertise and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.

[9]

R. Guo and B. Pei, Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson point processes, Stoch. Anal. Appl., 36 (2018), 751-766.  doi: 10.1080/07362994.2018.1461567.

[10]

X. Y. He, S. Han and J. Tao, Averaging principle for SDEs of neutral type driven by $G$-Brownian motion, Stoch. Dyn., 19 (2019), 1950004, 22 pp. doi: 10.1142/S0219493719500047.

[11]

M.-S. Hu and S.-G. Peng, On representation theorem of $G$-expectations and paths of $G$-Brownian motion, Acta Math. Appl. Sin., Eng. Ser., 25 (2009), 539-546.  doi: 10.1007/s10255-008-8831-1.

[12]

L. Y. HuY. Ren and T. B. 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.

[13]

G. J. Li and Q. G. Yang, Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by $G$-Brownian motion, Comput. Appl. Math., 37 (2018), 4301-4320.  doi: 10.1007/s40314-018-0581-y.

[14]

Q. Lin, Local time and Tanaka formula for the $G$-Brownian motion, J. Math. Anal. Appl., 398 (2013), 315-334.  doi: 10.1016/j.jmaa.2012.09.001.

[15]

Q. Lin, Some properties of stochastic differential equations driven by $G$-Brownian motion, Acta Math. Appl. Sin. Eng. Ser., 29 (2013), 923-942.  doi: 10.1007/s10114-013-0701-y.

[16]

X. R. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optimiz., 12 (1991), 253-533.  doi: 10.1080/01630569108816448.

[17]

X. R. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Ⅱ, Numer. Funct. Anal. Optimiz., 15 (1994), 65-76.  doi: 10.1080/01630569408816550.

[18]

X. R. Mao, Approximate solutions for stochastic differntial equations with pathwise uniqueness, Stoch. Anal. Appl., 12 (1994), 355-367.  doi: 10.1080/07362999408809356.

[19]

X. R. Mao, Stochastic Differential Equations and Applications, 2nd edition, Ellis Horwood, Chichester, 2007.

[20]

W. Mao and X. R. Mao, An averaging principle for neutral stochastic differential equations driven by Poisson random measure, Adv. Differ. Equ., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0802-x.

[21]

S. Peng, $G$-Expectation, $G$-Brownian motion and related stochastic calculus of Itô type, in Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541–567.

[22]

S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834.

[23]

Y. RenQ. 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.

[24]

Y. Ren and L. Y. Hu, A note on the stochastic differential equations driven by $G$-Brownian motion, Stat. Prob. Lett., 81 (2011), 580-585.  doi: 10.1016/j.spl.2011.01.010.

[25]

Y. RenX. J. Jia and L. Y. Hu, Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2157-2169.  doi: 10.3934/dcdsb.2015.20.2157.

[26]

Y. RenS. P. Lu and N. M. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinte delay, J. Comput. Appl. Math., 220 (2008), 364-372.  doi: 10.1016/j.cam.2007.08.022.

[27]

Y. Ren and N. M. Xia, A note on the Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinte delay, Appl. Math. Comput., 214 (2009), 457-461.  doi: 10.1016/j.amc.2009.04.013.

[28]

Y. Ren and N. M. Xia, Existence, uniqueness and stability of solutions to neutral stochastic functional differential equations with infinte delay, Appl. Math. Comput., 210 (2009), 72-79.  doi: 10.1016/j.amc.2008.11.009.

[29]

Y. S. Song, Uniqueness of the representation for $G$-martingales with finite variation, Electron. J. Prob., 17 (2012), 15 pp. doi: 10.1214/EJP.v17-1890.

[30]

L. Tan and D. Y. Lei, The averaging method for stochastic differential delay equations under non-Lipschitz conditions, Adv. Differ. Equ., 38 (2013), 12 pp. doi: 10.1186/1687-1847-2013-38.

[31]

F. Y. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331 (2007), 516-531.  doi: 10.1016/j.jmaa.2006.09.020.

[32]

J. Xu and B. Zhang, Martingale characterization of $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 232-248.  doi: 10.1016/j.spa.2008.02.001.

[33]

Y. XuJ. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.  doi: 10.1016/j.physd.2011.06.001.

[34]

Y. XuB. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2257-2267.  doi: 10.3934/dcdsb.2015.20.2257.

[35]

Y. Xu, B. Pei and Y. G. Li, An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal., 2014 (2014), Art. ID 479195, 10 pp. doi: 10.1155/2014/479195.

[36]

Y. XuB. Pei and Y. G. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38 (2015), 2120-2131.  doi: 10.1002/mma.3208.

[37]

Y. Xu, B. Pei and J.-L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dyn., 17 (2017), 1750013, 16 pp. doi: 10.1142/S0219493717500137.

[38]

B. ZhangJ. Xu and D. Kannan, Extension and application of Itô's formula under $G$-framework, Stoch. Anal. Appl., 28 (2010), 322-349.  doi: 10.1080/07362990903546595.

[39]

D. F. Zhang and Z. J. Chen, Exponential stability for stochastic differential equation driven by $G$-Brownian motion, Appl. Math. Lett., 25 (2012), 1906-1910.  doi: 10.1016/j.aml.2012.02.063.

[40]

M. Zhang and G. F. Zong, Almost periodic solutions for stochastic differential equations driven by $G$-Brownian motion, Comm. Stat. Theor. Meth., 44 (2015), 2371-2384.  doi: 10.1080/03610926.2013.863935.

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