THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

MILSTEIN-TYPE SEMI-IMPLICIT SPLIT-STEP NUMERICAL METHODS FOR NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS 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
DOI REFERENCE: https://doi.org/10.2298/TSCI180912325I
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 1, PAGES [S1 - S12]
REFERENCES
  1. 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, 181 (2002), 1, pp. 45-67
  2. Higham, D. J., et al., Strong Convergence of Euler Type Methods for Nonlinear Stochastic Differential Equations, SIAM J. Numer. Anal., 40 (2002), 3, pp. 1041-1063
  3. Schurz, H., An Axiomatic Approach to Numerical Approximations of Stochastic Processes, Int. J. of Num. Analysis and Modeling,3 (2006), 2, pp. 459-480
  4. 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, Mathematical, Physical, and Engineering Sciences, 467 (2011), 2130, pp. 1563-1576
  5. Hutzenthaler, M., et al., Strong Convergence of an Explicit Numerical Method for SDES with Non-Glob-ally Lipschitz Continuous Coefficients, Ann. Appl. Prob., 22 (2012), 4, pp. 1611-1641
  6. Mao, X., Szpruch, L., Strong Convergence and Stability of Implicit Numerical Methods for Stochas-tic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, J. of Comp. and Appl. Math., 238 (2013), Jan., pp. 14-28
  7. Liu, W., Mao, X., Strong Convergence of the Stopped Euler-Maruyama Method for Nonlinear Stochastic Differential Equations, Applied Math. and Comp., 223 (2013), Oct., pp 389-400
  8. Hutzenthaler, M., Jentzen, A., Numerical Approximations of Nonlinear Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, Memoirs of the Amer. Math. Soc., 236 (2015), 1112, pp. 1-104
  9. Izgi, B., Cetin, C., Some Moment Estimates for New Semi-Implicit Split-Step Methods, AIP Conference Proceedings, 1833 (2017), 1, 020041
  10. Izgi, B., Cetin, 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), Dec., pp. 62-79
  11. Mao, X., The Truncated Euler-Maruyama Method for Stochastic Differential Equations, J. of Comp. and Appl. Math., 290, (2015), Dec., pp. 370-384
  12. Wang, P., Li, Y., Split-Step Forward Methods for Stochastic Differential Equations, J. of Comp. and Appl. Math., 233 (2010), 10, pp. 2641-2651
  13. Wang, X., Gan, S., The Tamed Milstein Method for Commutative Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, J. of Differ. Equ. and Appl., 19 (2013), 3, pp. 466-490
  14. Guo, Q., et al., The Truncated Milstein Method for Stochastic Differential Equations with Commutative Noise, J. of Comp. and Appl. Math., 338 (2018), Aug., pp. 298-310
  15. Ashino, R., et al., Behind and Beyond the Matlab ODE Suite, Computers and Mathematics with Applica-tions, 40 (2000), 4-5, pp. 491-512

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence