Numerical method of highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. (English) Zbl 1486.65008
Summary: In this paper, we establish a partially truncated Euler-Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 34K50 | Stochastic functional-differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
Keywords:
partially truncated Euler-Maruyama method; neutral stochastic differential delay equations; Markovian switching; highly nonlinear and nonautonomous equationsReferences:
| [1] | Anderson, D. F.; Higham, D. J.; Sun, Y., Multilevel Monte Carlo for stochastic differential equations with small noise, SIAM J. Numer. Anal., 54, 505-529 (2016) · Zbl 1334.60128 · doi:10.1137/15M1024664 |
| [2] | Appleby, J. A.D.; Guzowska, M.; Kelly, C.; Rodkina, A., Preserving positivity in solutions of discretised stochastic differential equations, Appl. Math. Comput., 217, 763-774 (2010) · Zbl 1208.65013 |
| [3] | Arnold, L., Stochastic Differential Equations, Theory and Applications (1974), New York: Wiley, New York · Zbl 0278.60039 |
| [4] | Burrage, K.; Tian, T., Predictor-corrector methods of Runge-Kutta type for stochastic differential equations, SIAM J. Numer. Anal., 40, 1516-1537 (2002) · Zbl 1026.60075 · doi:10.1137/S0036142900372677 |
| [5] | Cong, Y.; Zhan, W.; Guo, Q., The partially truncated Euler-Maruyama method for highly nonlinear stochastic delay differential equations with Markovian switching, Int. J. Comput. Methods (2019) · Zbl 07205469 · doi:10.1142/S0219876219500142 |
| [6] | Guo, Q.; Liu, W.; Mao, X.; Yue, R., The partially truncated Euler-Maruyama method and its stability and boundedness, Appl. Numer. Math., 115, 235-251 (2017) · Zbl 1358.65008 · doi:10.1016/j.apnum.2017.01.010 |
| [7] | Guo, Q.; Mao, X.; Yue, R., The truncated Euler-Maruyama method for stochastic differential delay equations, Numer. Algorithms, 78, 599-624 (2018) · Zbl 1414.60043 · doi:10.1007/s11075-017-0391-0 |
| [8] | Higham, D. J.; Mao, X.; Stuart, A. M., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40, 1041-1063 (2002) · Zbl 1026.65003 · doi:10.1137/S0036142901389530 |
| [9] | Hutzenthaler, M.; Jentzen, A.; Kloeden, P. E., Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A, 467, 1563-1576 (2010) · Zbl 1228.65014 · doi:10.1098/rspa.2010.0348 |
| [10] | Hutzenthaler, M.; Jentzen, A.; Kloeden, P. E., Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22, 1611-1641 (2012) · Zbl 1256.65003 · doi:10.1214/11-AAP803 |
| [11] | Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Berlin: Springer, Berlin · Zbl 0925.65261 · doi:10.1007/978-3-662-12616-5 |
| [12] | Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., Neutral stochastic differential delay equation with Markovian switching, Stoch. Anal. Appl., 21, 819-847 (2003) · Zbl 1025.60028 · doi:10.1081/SAP-120022865 |
| [13] | Lan, G., 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 |
| [14] | Lan, G.; Yuan, C., 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 |
| [15] | Li, X.; Mao, X., A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48, 2329-2334 (2012) · Zbl 1257.93106 · doi:10.1016/j.automatica.2012.06.045 |
| [16] | Li, X.; Mao, X.; Yin, G., 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, 847-892 (2019) · Zbl 1461.65007 · doi:10.1093/imanum/dry015 |
| [17] | Liu, W.; Foondun, M.; Mao, X., Mean square polynomial stability of numerical solutions to a class of stochastic differential equations, Stat. Probab. Lett., 92, 173-182 (2014) · Zbl 1294.60094 · doi:10.1016/j.spl.2014.06.002 |
| [18] | Liu, W.; Mao, X., Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput., 223, 389-400 (2013) · Zbl 1329.65018 |
| [19] | Liu, W.; Mao, X.; Tang, J.; Wu, Y., Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations, Appl. Numer. Math., 153, 66-81 (2020) · Zbl 1456.65007 · doi:10.1016/j.apnum.2020.02.007 |
| [20] | Mao, X., Stochastic Differential Equations and Applications (2007), Chichester: Horwood, Chichester · Zbl 1138.60005 |
| [21] | Mao, X., 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 |
| [22] | Mao, X., Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296, 362-375 (2016) · Zbl 1378.65036 · doi:10.1016/j.cam.2015.09.035 |
| [23] | Mao, X.; Shen, Y., Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stoch. Process. Appl., 118, 1385-1406 (2008) · Zbl 1143.60041 · doi:10.1016/j.spa.2007.09.005 |
| [24] | Mao, X.; Yuan, C., Stochastic Differential Equations with Markovian Switching (2006), London: Imperial College Press, London · Zbl 1126.60002 · doi:10.1142/p473 |
| [25] | Mao, X.; Yuan, C.; Yin, G., Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions, J. Comput. Appl. Math., 205, 936-948 (2007) · Zbl 1121.65011 · doi:10.1016/j.cam.2006.01.052 |
| [26] | Milstein, G. N.; Platen, E.; Schurz, H., Balanced implicit methods for stiff stochastic system, SIAM J. Numer. Anal., 35, 1010-1019 (1998) · Zbl 0914.65143 · doi:10.1137/S0036142994273525 |
| [27] | Øksendal, B., Stochastic Differential Equations: An Introduction with Applications (2000), Berlin: Springer, Berlin · Zbl 0567.60055 |
| [28] | Sabanis, S., A note on tamed Euler approximations, Electron. Commun. Probab., 18 (2013) · Zbl 1329.60237 · doi:10.1214/ECP.v18-2824 |
| [29] | Sabanis, S., Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients, Ann. Appl. Probab., 26, 2083-2105 (2016) · Zbl 1352.60101 · doi:10.1214/15-AAP1140 |
| [30] | Saito, Y.; Mitsui, T., T-stability of numerical scheme for stochastic differential equations, World Sci. Ser. Appl. Anal., 2, 333-344 (1993) · Zbl 0834.65146 |
| [31] | Skorohod, A. V., Asymptotic Methods in the Theory of Stochastic Differential Equations (1989), Providence: Am. Math. Soc., Providence · Zbl 0695.60055 |
| [32] | Szpruch, L.; Mao, X.; Higham, D.; Pan, J., Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numer. Math., 51, 405-425 (2011) · Zbl 1230.65011 · doi:10.1007/s10543-010-0288-y |
| [33] | Tan, L.; Yuan, C., Convergence rates of theta-method for NSDDEs under non-globally Lipschitz continuous coefficients, Bull. Math. Sci., 9, 3231-3243 (2019) · Zbl 07365323 · doi:10.1142/S1664360719500061 |
| [34] | Wu, F.; Mao, X., Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46, 1821-1841 (2008) · Zbl 1173.65004 · doi:10.1137/070697021 |
| [35] | Wu, F.; Mao, X.; Chen, K., The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Appl. Numer. Math., 59, 2641-2658 (2009) · Zbl 1172.65009 · doi:10.1016/j.apnum.2009.03.004 |
| [36] | Wu, F.; Mao, X.; Szpruch, L., Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math., 115, 681-697 (2010) · Zbl 1193.65009 · doi:10.1007/s00211-010-0294-7 |
| [37] | Yuan, C.; Glover, W., Approximate solutions of stochastic differential delay equations with Markovian switching, J. Comput. Appl. Math., 194, 207-226 (2006) · Zbl 1098.65005 · doi:10.1016/j.cam.2005.07.004 |
| [38] | Zhang, W.; Song, M.; Liu, M., 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 |
| [39] | Zhou, S.; Wu, F., Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching, J. Comput. Appl. Math., 229, 85-96 (2009) · Zbl 1168.65007 · doi:10.1016/j.cam.2008.10.013 |
| [40] | Zong, X.; Wu, F.; Huang, C., 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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
