Strong convergence and asymptotic exponential stability of modified truncated EM method for neutral stochastic differential equations with time-dependent delay. (English) Zbl 1536.65009
Summary: In this paper, we consider asymptotic exponential stability of the exact solution and the corresponding modified truncated EM (MTEM) method for neutral stochastic differential equations (NSDEs) with time-dependent delay. We obtain sufficient conditions under which the MTEM method replicates the exponential stability of the exact solution no matter time-dependent delay \(\delta (t)\) is bounded or not. To make sure that the stability conclusions are meaningful, we will obtain the strong convergence of the MTEM method at first. Two examples are presented to illustrate our conclusions.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34K50 | Stochastic functional-differential equations |
| 34K40 | Neutral functional-differential equations |
References:
| [1] | Gan, SQ; Schurz, H.; Zhang, HM, Mean-square convergence of stochastic θ-methods for nonlinear neutral stochastic differential delay equations, Int. J. Numer. Anal. Model. Ser. B, 8, 2, 201-213 (2011) · Zbl 1215.65014 |
| [2] | Hu, YZ; Wu, FK; Huang, CM, General decay pathwise stability of neutral stochastic differential equations with unbounded delay, Acta Math. Sin. (Engl. Ser.), 27, 11, 2153-2168 (2011) · Zbl 1227.60074 · doi:10.1007/s10114-011-9456-5 |
| [3] | Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), Amsterdam: North-Holland Publishing Co., Amsterdam · Zbl 0495.60005 · doi:10.1016/S0924-6509(08)70221-6 |
| [4] | Kloeden, PE; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Berlin: Springer-Verlag, Berlin · Zbl 0752.60043 · doi:10.1007/978-3-662-12616-5 |
| [5] | Lakshmikantham, V.; Wen, LZ; Zhang, BG, Theory of Differential Equations with Unbounded Delay (1994), Dordrecht: Kluwer Academic Publishers Group, Dordrecht · Zbl 0823.34069 · doi:10.1007/978-1-4615-2606-3 |
| [6] | Lan, GQ, Asymptotic exponential stability of modified truncated EM method for neutral stochastic differential delay equations, J. Comput. Appl. Math., 340, 334-341 (2018) · Zbl 1391.65021 · doi:10.1016/j.cam.2018.03.001 |
| [7] | Lan, GQ; Wang, QS, Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations, J. Comput. Appl. Math., 362, 83-98 (2019) · Zbl 1418.60059 · doi:10.1016/j.cam.2019.05.021 |
| [8] | Lan, GQ; Xia, F., Strong convergence rates of modified truncated EM method for stochastic differential equations, J. Comput. Appl. Math., 334, 1-17 (2018) · Zbl 1378.60083 · doi:10.1016/j.cam.2017.11.024 |
| [9] | Lan, GQ; Xia, F.; Wang, QS, Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay, J. Comput. Appl. Math., 346, 340-356 (2019) · Zbl 1403.65005 · doi:10.1016/j.cam.2018.07.024 |
| [10] | Lan, GQ; Yuan, CG, Exponential stability of the exact solutions and θ-EM approximations to neutral SDDEs with Markov switching, J. Comput. Appl. Math., 285, 230-242 (2015) · Zbl 1319.60128 · doi:10.1016/j.cam.2015.02.028 |
| [11] | Liu, LN; Zhu, QX, Mean-square stability of two classes of theta method for neutral stochastic differential delay equations, J. Comput. Appl. Math., 305, 55-67 (2016) · Zbl 1386.65039 · doi:10.1016/j.cam.2016.03.021 |
| [12] | Mao, XR, Stochastic Differential Equations and Applications (2007), Chichester: Horwood Publishing Limited, Chichester · Zbl 1138.60005 |
| [13] | Mao, XR, The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290, 370-384 (2015) · Zbl 1330.65016 · doi:10.1016/j.cam.2015.06.002 |
| [14] | Mao, XR; Yuan, CG, Stochastic Differential Equations with Markovian Switching (2006), London: Imperial College Press, London · Zbl 1126.60002 · doi:10.1142/p473 |
| [15] | Milosevic, M., Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method, Math. Comput. Modelling, 54, 9-10, 2235-2251 (2011) · Zbl 1235.65009 · doi:10.1016/j.mcm.2011.05.033 |
| [16] | Milošević, M., Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation, Math. Comput. Modelling, 57, 3-4, 887-899 (2013) · Zbl 1305.34138 · doi:10.1016/j.mcm.2012.09.016 |
| [17] | Mo, HY; Zhao, XY; Deng, FQ, Mean-square stability of the backward Euler-Maruyama method for neutral stochastic delay differential equations with jumps, Math. Methods Appl. Sci., 40, 5, 1794-1803 (2017) · Zbl 1360.60113 · doi:10.1002/mma.4098 |
| [18] | Obradović, M.; Milošević, M., Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler-Maruyama method, J. Comput. Appl. Math., 309, 244-266 (2017) · Zbl 1351.60074 · doi:10.1016/j.cam.2016.06.038 |
| [19] | Tan L., Yuan C.G., Convergence rates of theta-method for neutral SDDEs under non-globally Lipschitz continuous coefficients. 2017, arXiv:1701.00223 |
| [20] | Tan, L.; Yuan, CG, Strong convergence of a tamed theta scheme for NSDDEs with onesided Lipschitz drift, Appl. Math. Comput., 338, 607-623 (2018) · Zbl 1427.65010 |
| [21] | Wang, WQ; Chen, YP, Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations, Appl. Numer. Math., 61, 5, 696-701 (2011) · Zbl 1210.65014 · doi:10.1016/j.apnum.2011.01.003 |
| [22] | Wu, FK; Hu, SG; Mao, XR, Razumikhin-type theorem for neutral stochastic functional differential equations with unbounded delay, Acta Math. Sci. Ser. B Engl. Ed., 31, 4, 1245-1258 (2011) · Zbl 1249.34235 |
| [23] | Yue C., Zhao L.B., Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient. J. Comput. Appl. Math., 2021, 382: Paper No. 113087, 17 pp. · Zbl 1484.65019 |
| [24] | Zhang, HM; Gan, SQ, Mean-square convergence of one-step methods for neutral stochastic differential delay equations, Appl. Math. Comput., 204, 2, 884-890 (2008) · Zbl 1155.65010 |
| [25] | Zhang, W.; Song, MH; Liu, MZ, Strong convergence of the partially truncated Euler-Maruyama method for a class of stochastic differential delay equations, J. Comput. Appl. Math., 335, 114-128 (2018) · Zbl 1444.34099 · doi:10.1016/j.cam.2017.11.030 |
| [26] | Zhao, JJ; Yi, YL; Xu, Y., Strong convergence and stability of the split-step theta method for highly nonlinear neutral stochastic delay integro differential equation, Appl. Numer. Math., 157, 385-404 (2020) · Zbl 1474.45082 · doi:10.1016/j.apnum.2020.06.013 |
| [27] | Zhu, QX, Stability analysis of stochastic delay differential equations with Levy noise, Systems Control Lett., 118, 62-68 (2018) · Zbl 1402.93260 · doi:10.1016/j.sysconle.2018.05.015 |
| [28] | Zong, XF; Wu, FK; Huang, CM, Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations, J. Comput. Appl. Math., 286, 172-185 (2015) · Zbl 1320.34111 · doi:10.1016/j.cam.2015.03.016 |
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