Abstract
In this paper, we construct a two-stage truncated Runge–Kutta (TSRK2) method for highly nonlinear stochastic differential equations (SDEs) with non-global Lipschitz coefficients. TSRK2 is an explicit method and includes some free parameters that can extend the accuracy of the results and stability regions. We show that this method can achieve a strong convergence rate arbitrarily close to half under local Lipschitz and Khasiminskii conditions. We study the mean square stability properties (MS-stability) of the method based on a scalar linear test equation with multiplicative noise, and the advantages of our results are highlighted by comparing them with those of the truncated Euler–Maruyama method. We also analyze the asymptotic stability properties of the method. We show that the proposed method can preserve the asymptotic stability of the original system under mild conditions. Finally, we report some numerical experiments to illustrate the effectiveness of the proposed method. As a result, we show that the new method has good properties not only in terms of practical errors but also in terms of stability.
Similar content being viewed by others
References
Buckwar, E., Samson, A., Tamborrino, M., Tubikanec, I.: A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model. Appl. Numer. Math. 179, 191–220 (2022)
Burrage, K., Burrage, P.M., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. Lond. A. 460, 373–402 (2004)
Gan, S., He, Y., Wang, X.: Tamed Runge–Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients. Appl. Numer. Mathe. 152, 379–402 (2020)
Gillespie, D.T.: The chemical Langevin equation. J. Chem. Phys. 113(1), 297–306 (2000)
Guo, Q., Liu, W., Mao, X.: A note on the partially truncated Euler–Maruyama method. Appl. Numer. Math. 130, 157–170 (2018)
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)
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)
Haghighi, A., Hosseini, S.M., Rößler, A.: Diagonally drift-implicit Runge–Kutta methods of strong order one for stiff stochastic differential systems. J. Comput. Appl. Math. 293, 82–93 (2016)
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)
Hu, L., Li, X., Mao, X.: Convergence rate and stability of the truncated Euler–Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 337, 274–289 (2018)
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. Lond. Ser. A Math. Phys. Eng. Sci. 467, 1563–1576 (2011)
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Komori, Y., Burrage, K.: Strong first order S-ROCK methods for stochastic differential equations. J. Comput. Appl. Math. 242, 261–274 (2013)
Lewis, A.L.: Option Valuation Under Stochastic Volatility: With Mathematica Code. Finance Press, California (2000)
Li, X., Yin, G.: Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: convergence and its rate. J. Comput. Appl. Math. 374, 112,771 (2020)
Liu, W., Mao, X.: Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations. Appl. Math. Comput. 223, 389–400 (2013)
Mao, X.: A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268(1), 125–142 (2002)
Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood, Chichester (2007)
Mao, X.: The truncated Euler–Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 290, 370–384 (2015)
Mao, X.: Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 296, 362–375 (2016)
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)
Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer Academic, Dordrecht (1995)
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)
Nouri, K.: Improving split-step forward methods by ODE solver for stiff stochastic differential equations. Math. Sci. 16(1), 51–57 (2022)
Nouri, K., Ranjbar, H., Torkzadeh, L.: Improved Euler–Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15(3), 140 (2018)
Øksendal, B.: Stochastic Differential Equations. An Introduction With Applications. In: Universitext, 6th edn. Springer-Verlag, Berlin (2003)
Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Math. Anal. 48(3), 922–952 (2010)
Sabanis, S.: A note on tamed Euler approximations. Electron. Commun. Probab. 18, 1–10 (2013)
Tian, T., Burrage, K.: Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math. 38(1), 167–185 (2001)
Torkzadeh, L.: Mean-square convergence analysis of the semi-implicit scheme for stochastic differential equations driven by the Wiener processes. Math. Sci. (2021). https://doi.org/10.1007/s40096-021-00440-2
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)
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)
Yang, H., Huang, J.: Convergence and stability of modified partially truncated Euler–Maruyama method for nonlinear stochastic differential equations with hölder continuous diffusion coefficient. J. Comput. Appl. Math. 404, 113, 895 (2022)
Yang, H., Wu, F., Kloeden, P.E., Mao, X.: The truncated Euler–Maruyama method for stochastic differential equations with hölder diffusion coefficients. J. Comput. Appl. Math. 366, 112, 379 (2020)
Acknowledgements
The author would like to thank the reviewers for their constructive comments, which helped to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Haghighi, A. An explicit two-stage truncated Runge–Kutta method for nonlinear stochastic differential equations. Math Sci (2023). https://doi.org/10.1007/s40096-023-00508-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40096-023-00508-1