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Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations
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  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 & 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.  Google Scholar [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.  Google Scholar [3] M. Abouagwa, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. Denis, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] L. Y. Hu, Y. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] X. R. Mao, Approximate solutions for stochastic differntial equations with pathwise uniqueness, Stoch. Anal. Appl., 12 (1994), 355-367.  doi: 10.1080/07362999408809356.  Google Scholar [19] X. R. Mao, Stochastic Differential Equations and Applications, 2nd edition, Ellis Horwood, Chichester, 2007. Google Scholar [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.  Google Scholar [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. Google Scholar [22] S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834. Google Scholar [23] Y. Ren, Q. 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.  Google Scholar [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.  Google Scholar [25] Y. Ren, X. 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.  Google Scholar [26] Y. Ren, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] Y. Xu, J. 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.  Google Scholar [34] Y. Xu, B. 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.  Google Scholar [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.  Google Scholar [36] Y. Xu, B. 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.  Google Scholar [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.  Google Scholar [38] B. Zhang, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] M. Abouagwa, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. Denis, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] L. Y. Hu, Y. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] X. R. Mao, Approximate solutions for stochastic differntial equations with pathwise uniqueness, Stoch. Anal. Appl., 12 (1994), 355-367.  doi: 10.1080/07362999408809356.  Google Scholar [19] X. R. Mao, Stochastic Differential Equations and Applications, 2nd edition, Ellis Horwood, Chichester, 2007. Google Scholar [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.  Google Scholar [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. Google Scholar [22] S. Peng, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, preprint, arXiv: 0711.2834. Google Scholar [23] Y. Ren, Q. 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.  Google Scholar [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.  Google Scholar [25] Y. Ren, X. 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.  Google Scholar [26] Y. Ren, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] Y. Xu, J. 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.  Google Scholar [34] Y. Xu, B. 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.  Google Scholar [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.  Google Scholar [36] Y. Xu, B. 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.  Google Scholar [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.  Google Scholar [38] B. Zhang, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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