Abstract
Motivated by truncated Euler–Maruyama (EM) method established by Mao (J Comput Appl Math 290:370–384, 2015, J Comput Appl Math 296:362–375, 2016), a state-of-the-art scheme named full-implicit truncated EM method is derived in this paper, aiming to use shorter runtime and larger stepsize, as well as to obtain better stability property. Weaker restrictions on truncated functions of full-implicit truncated EM scheme have been obtained, which solve the disadvantage of Mao (2015) requiring the stepsize to be so small that sometimes the truncated EM would be inapplicable. The superiority of our results will be highlighted by the comparisons with the achievements in Mao (2015, 2016) as well as others on the implicit Euler scheme and semi-implicit truncated EM scheme. Numerical examples verify the order of \(L^q\)-convergence, cheaper computational costs, wider stepsize and better stability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht (1955)
Maruyama, G.: Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo. 4(1), 48–90 (1955)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (2007)
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. 467, 1563–1576 (2011)
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)
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(4), 1611–1641 (2012)
Hutzenthaler, M., Jentzen, A.: Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Found. Comput. Math. 11(6), 657–706 (2011)
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)
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.: A note on the partially truncated Euler–Maruyama method. J Appl. Numer. Math. 130, 157–170 (2018)
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)
Li, Q., Gan, S.: Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. J. Appl. Math. Comput. 39, 69–87 (2012)
Guo, Q., Mao, X., Yue, R.: The truncated Euler–Maruyama method for stochastic differential delay equations. Numer. Algorithm 78, 599–624 (2018)
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)
Lan, G., Xia, F.: Strong convergence rates of modified truncated EM method for stochastic differential equations. J. Comput. Appl. Math. 334, 1–17 (2018)
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)
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(1), 14–28 (2013)
Mao, X., Szpruch, L.: Strong convergence rates for backward Euler–Maruyama method for nonlinear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics 85, 144–171 (2013)
Bastani, A.F., Tahmasebi, M.: Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift. J. Comput. Appl. Math. 236, 1903–1918 (2012)
Wang, X., Gan, S., Wang, D.: A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise. BIT Numer. Math. 52(3), 741–772 (2012)
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)
Acknowledgements
The authors are grateful to the anonymous referees for reading of a preliminary version of the manuscript. Their valuable suggestions significantly improve the quality of this paper. And the authors would like to thank A. Prof. Feiyan Xiao for her previously helpful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Wen, H. Convergence rates of full-implicit truncated Euler–Maruyama method for stochastic differential equations. J. Appl. Math. Comput. 60, 147–168 (2019). https://doi.org/10.1007/s12190-018-1206-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-018-1206-8