Split-step theta Milstein methods for SDEs with non-globally Lipschitz diffusion coefficients. (English) Zbl 1492.65028
Summary: The present work aims to analyze mean-square convergence rates of split-step theta Milstein methods with method parameters \(\theta\in[\frac{1}{2},1]\) for stochastic differential equations with non-globally Lipschitz diffusion coefficients. The expected convergence rate of order one is successfully proved under more relaxed conditions, compared to existing relevant results. Numerical examples are also offered to confirm the theoretical findings.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
Keywords:
stochastic differential equations; non-globally Lipschitz coefficients; split-step theta Milstein methods; strong convergence ratesReferences:
| [1] | Andersson, A.; Kruse, R., Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition, BIT Numer. Math., 57, 1, 21-53 (2017) · Zbl 1365.65010 |
| [2] | Beyn, W.-J.; Isaak, E.; Kruse, R., Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput., 67, 3, 955-987 (2016) · Zbl 1362.65011 |
| [3] | Beyn, W.-J.; Isaak, E.; Kruse, R., Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput., 70, 3, 1042-1077 (2017) · Zbl 1397.65013 |
| [4] | Chassagneux, J.; Jacquier, A.; Mihaylov, I., An explicit Euler scheme with strong rate of convergence for financial SDEs with non-lipschitz coefficients, SIAM J. Financ. Math., 7, 1, 993-1021 (2016) · Zbl 1355.60072 |
| [5] | Fang, W.; Giles, M. B., Adaptive Euler-Maruyama method for SDEs with nonglobally Lipschitz drift, Ann. Appl. Probab., 30, 2, 526-560 (2020) · Zbl 1464.60061 |
| [6] | Gan, S.; He, Y.; Wang, X., Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients, Appl. Numer. Math., 152, 379-402 (2020) · Zbl 1441.65013 |
| [7] | Guo, Q.; Liu, W.; Mao, X.; Yue, R., The truncated Milstein method for stochastic differential equations with commutative noise, J. Comput. Appl. Math., 338, 298-310 (2018) · Zbl 1503.65014 |
| [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, 3, 1041-1063 (2002) · Zbl 1026.65003 |
| [9] | Higham, D. J.; Mao, X.; Szpruch, L., Convergence, non-negativity and stability of a new Milstein scheme with applications to finance, Discrete Contin. Dyn. Syst., Ser. B, 18, 8, 2083-2100 (2013) · Zbl 1279.60068 |
| [10] | Hu, Y., Semi-implicit euler-maruyama scheme for stiff stochastic equations, Stochastic Analysis and Related Topics V: The Silvri Workshop. Stochastic Analysis and Related Topics V: The Silvri Workshop, Prog. Probab., 38, 183-202 (1996) · Zbl 0848.60057 |
| [11] | Huang, C., Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math., 236, 16, 4016-4026 (2012) · Zbl 1251.65003 |
| [12] | Hutzenthaler, M.; Jentzen, A., Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Am. Math. Soc., 236, 1112 (2015) · Zbl 1330.60084 |
| [13] | Hutzenthaler, M.; Jentzen, A., On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients, Ann. Probab., 48, 1, 53-93 (2020) · Zbl 07206753 |
| [14] | 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, Math. Phys. Eng. Sci., 467, 2130, 1563-1576 (2011) · Zbl 1228.65014 |
| [15] | Hutzenthaler, M.; Jentzen, A.; Kloeden, P. E., Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz coefficients, Ann. Appl. Probab., 22, 4, 1611-1641 (2012) · Zbl 1256.65003 |
| [16] | Hutzenthaler, M.; Jentzen, A.; Wang, X., Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations, Math. Comput., 87, 311, 1353-1413 (2018) · Zbl 1432.65011 |
| [17] | Kelly, C.; Lord, G.; Sun, F., Strong convergence of an adaptive time-stepping Milstein method for SDEs with one-sided Lipschitz drift (2019), preprint |
| [18] | Kelly, C.; Lord, G. J., Adaptive time-stepping strategies for nonlinear stochastic systems, IMA J. Numer. Anal., 38, 3, 1523-1549 (2017) · Zbl 1477.65023 |
| [19] | Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations, Vol. 23 (1992), Springer Science & Business Media · Zbl 0925.65261 |
| [20] | Kumar, C.; Sabanis, S., On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients, BIT Numer. Math., 59, 4, 929-968 (2019) · Zbl 1505.65011 |
| [21] | Mao, X., Stochastic Differential Equations and Applications (2007), Elsevier · Zbl 1138.60005 |
| [22] | Mao, X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290, 370-384 (2015) · Zbl 1330.65016 |
| [23] | 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 |
| [24] | Mao, X.; Szpruch, L., Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238, 14-28 (2013) · Zbl 1262.65012 |
| [25] | Mao, X.; Szpruch, L., Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85, 1, 144-171 (2013) · Zbl 1304.65009 |
| [26] | Mattingly, J. C.; Stuart, A. M.; Higham, D. J., Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stoch. Process. Appl., 101, 2, 185-232 (2002) · Zbl 1075.60072 |
| [27] | Milstein, G. N.; Tretyakov, M. V., Stochastic Numerics for Mathematical Physics (2013), Springer Science & Business Media · Zbl 1280.35097 |
| [28] | Sabanis, S., A note on tamed Euler approximations, Electron. Commun. Probab., 18, 47, 1-10 (2013) · Zbl 1329.60237 |
| [29] | Sabanis, S., Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients, Ann. Appl. Probab., 26, 4, 2083-2105 (2016) · Zbl 1352.60101 |
| [30] | Tretyakov, M. V.; Zhang, Z., A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51, 6, 3135-3162 (2013) · Zbl 1293.60069 |
| [31] | Wang, X., Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients: applications to financial models (2020), preprint |
| [32] | Wang, X.; Gan, S., The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equ. Appl., 19, 3, 466-490 (2013) · Zbl 1262.65008 |
| [33] | Wang, X.; Wu, J.; Dong, B., Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition, BIT Numer. Math., 60, 3, 759-790 (2020) · Zbl 1469.65033 |
| [34] | X. Wu, S. Gan, Convergence rates of split-step theta methods for SDEs with non-globally Lipschitz diffusion coefficients, Preprint, 2021. |
| [35] | Yue, C., High-order split-step theta methods for non-autonomous stochastic differential equations with non-globally Lipschitz continuous coefficients, Math. Methods Appl. Sci., 39, 9, 2380-2400 (2016) · Zbl 1338.65010 |
| [36] | Yue, C.; Huang, C.; Jiang, F., Strong convergence of split-step theta methods for non-autonomous stochastic differential equations, Int. J. Comput. Math., 91, 10, 2260-2275 (2014) · Zbl 1309.65011 |
| [37] | Zhang, Z.; Ma, H., Order-preserving strong schemes for SDEs with locally Lipschitz coefficients, Appl. Numer. Math., 112, 1-16 (2017) · Zbl 1354.65017 |
| [38] | Zong, X.; Wu, F.; Xu, G., Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations, J. Comput. Appl. Math., 336, 8-29 (2018) · Zbl 1382.65026 |
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