## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Milstein-type semi-implicit split-step numerical methods for nonlinear SDE with locally Lipschitz drift terms

**ABSTRACT**

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of nonlinear stochastic differential equations (SDE) with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein and truncated Milstein procedures on nonlinear SDE including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

**KEYWORDS**

PAPER SUBMITTED: 2018-09-12

PAPER REVISED: 2018-10-01

PAPER ACCEPTED: 2018-10-26

PUBLISHED ONLINE: 2018-12-16

- Ashino, R., et al., Behind and Beyond the Matlab ODE Suite. CRM-2651, (2000)
- Deang, J., et al., Modeling and computation of random thermal fluctuations and material defects in the Ginzburg-Landau model for superconductivity, J. of Comp. Physics (2002), 181, pp. 45-67
- Guo, Q., et al., The truncated Milstein method for stochastic differential equations with commutative noise, J. of Comp. and Appl. Math., 338, (2018), pp. 298-310.
- Higham, D. J., et al., Strong convergence of Euler type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (3), (2002), pp. 1041-1063.
- Hutzenthaler, M., and Jentzen, A., Numerical approximations of nonlinear stochastic differential equations with non-globally Lipschitz continuous coefficients, Memoirs of the Amer. Math. Soc., (2015) 236, 1112.
- Hutzenthaler, M., et al., Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society, A, 467, 2130, (2011), pp. 1563-1576.
- Hutzenthaler, M., et al., Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients, Ann. Appl. Prob., 22, (2012), pp. 1611-1641.
- İzgi, B. and Çetin, C., Some moment estimates for new semi-implicit split-step methods, AIP Conference Proceedings, (2017), Vol. 1833. No.1. AIP Publishing.
- İzgi, B. and Çetin, C., Semi-implicit split-step numerical methods for a class of nonlinear SDEs with non-Lipschitz drift terms, J. of Comp. and Appl. Math., 343, (2018), pp. 62-79.
- İzgi, B. and Çetin, C., Strong convergence of semi-implicit split-step methods for SDE with locally Lipschitz coefficients, Working paper, 2018.
- İzgi, B. and Çetin, C., Strong convergence of Milstein-type semi-implicit split-step methods for SDEs with locally Lipschitz coefficients, Working paper, 2018.
- Liu, W., and Mao, X., Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Applied Math and Comp., 223, (2013), pp 389-400.
- Mao, X., The truncated Euler-Maruyama method for stochastic differential equations, J. of Comp. and Appl. Math., 290, (2015), pp. 370-384.
- Mao, X., and Szpruch, L., Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. of Comp. and Appl. Math., 238, (2013), pp. 14-28.
- Schurz, H., An axiomatic approach to numerical approximations of stochastic processes, Int. J. of Num. Analysis and Modeling, 3 (02), (2006), pp. 459-480.
- Wang, X., and Gan, S., The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. of Differ. Equ. and Appl., Vol. 19, No. 3, (2013), pp. 466-490.
- Wang, P., and Li, Y., Split-step forward methods for stochastic differential equations, J. of Comp. and Appl. Math., 233, (2010), pp. 2641-2651.