Acknowledgement
The authors are deeply grateful to the anonymous referees and the editor for their careful reading, valuable comments and correction of some errors, which helped to improve the paper a lot.
References
- X. Bai and Y. 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. Engl. Ser. 30 (2014), no. 3, 589-610. https://doi.org/10.1007/s10255-014-0405-9
- J. Bao and X. Huang, Approximations of McKean-Vlasov stochastic differential equations with irregular coefficients, J. Theoret. Probab. 35 (2022), no. 2, 1187-1215. https://doi.org/10.1007/s10959-021-01082-9
- J. Bao, X. Huang, and C. Yuan, Convergence rate of Euler-Maruyama scheme for SDEs with Holder-Dini continuous drifts, J. Theoret. Probab. 32 (2019), no. 2, 848-871. https://doi.org/10.1007/s10959-018-0854-9
- J. Bao and C. Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3231-3243. https://doi.org/10.1090/S0002-9939-2013-11886-1
- S. Deng, C. Fei, W. Fei, and X. Mao, Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method, Appl. Math. Lett. 96 (2019), 138-146. https://doi.org/10.1016/j.aml.2019.04.022
- L. Denis, M. Hu, and S. G. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal. 34 (2011), no. 2, 139-161. https://doi.org/10.1007/s11118-010-9185-x
- F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl. 119 (2009), no. 10, 3356-3382. https://doi.org/10.1016/j.spa.2009.05.010
- D. J. Higham, X. Mao, and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 1041-1063. https://doi.org/10.1137/S0036142901389530
- M. Hu and S. Peng, On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 539-546. https://doi.org/10.1007/s10255-008-8831-1
- M. Hutzenthaler, A. Jentzen, and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611-1641. https://doi.org/10.1214/11-AAP803
- P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), 23, Springer, Berlin, 1992. https://doi.org/10.1007/978-3-662-12616-5
- 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), no. 1, 235-255. https://doi.org/10.1016/j.jmaa.2016.02.042
- X. Li, X. Mao, and G. Yin, Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment and stability, IMA J. Numer. Anal. 39 (2019), no. 2, 847-892. https://doi.org/10.1093/imanum/dry015
- X. Li and S. Peng, Stopping times and related Ito's calculus with G-Brownian motion, Stochastic Process. Appl. 121 (2011), no. 7, 1492-1508. https://doi.org/10.1016/j.spa.2011.03.009
- G. Li and Q. Yang, Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by G-Brownian motion, Comput. Appl. Math. 37 (2018), no. 4, 4301-4320. https://doi.org/10.1007/s40314-018-0581-y
- X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood, Chichester, 1997.
- X. Mao, The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370-384. https://doi.org/10.1016/j.cam.2015.06.002
- S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Ito type, in Stochastic analysis and applications, 541-567, Abel Symp., 2, Springer, Berlin, 2007. https://doi.org/10.1007/978-3-540-70847-6_25
- S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl. 118 (2008), no. 12, 2223-2253. https://doi.org/10.1016/j.spa.2007.10.015
- S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, Probability Theory and Stochastic Modelling, 95, Springer, Berlin, 2019. https://doi.org/10.1007/978-3-662-59903-7
- H. M. Soner, N. Touzi, and J. Zhang, Martingale representation theorem for the G-expectation, Stochastic Process. Appl. 121 (2011), no. 2, 265-287. https://doi.org/10.1016/j.spa.2010.10.006
- Y. Song, Some properties on G-evaluation and its applications to G-martingale decomposition, Sci. China Math. 54 (2011), no. 2, 287-300. https://doi.org/10.1007/s11425-010-4162-9
- Y. Song, Uniqueness of the representation for G-martingales with finite variation, Electron. J. Probab. 17 (2012), no. 24, 15 pp. https://doi.org/10.1214/EJP.v17-1890
- R. Ullah and F. Faizullah, On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion, Eur. Phys. J. Plus (2017), 132: 435 https://doi.org/10.1140/epjp/i2017-11700-9
- H. Wu, J. Hu, and C. Yuan, Stability of numerical solution to pantograph stochastic functional differential equations, Appl. Math. Comput. 431 (2022), Paper No. 127326, 13 pp. https://doi.org/10.1016/j.amc.2022.127326
- J. Yang and W. Zhao, Numerical simulations for G-Brownian motion, Front. Math. China 11 (2016), no. 6, 1625-1643. https://doi.org/10.1007/s11464-016-0504-9
- C. Yuan and X. Mao, A note on the rate of convergence of the Euler-Maruyama method for stochastic differential equations, Stoch. Anal. Appl. 26 (2008), no. 2, 325-333. https://doi.org/10.1080/07362990701857251