Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T06:36:22.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence rate of the EM algorithm for SDEs with low regular drifts

Part of: Stochastic analysis Functional-differential and differential-difference equations

Published online by Cambridge University Press:  14 February 2022

Jianhai Bao ,
Xing Huang  and
Shao-Qin Zhang
Jianhai Bao*
Affiliation:
Tianjin University
Xing Huang*
Affiliation:
Tianjin University
Shao-Qin Zhang*
Affiliation:
Central University of Finance and Economics
*
*Postal address: Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.
*Postal address: Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.
****Postal address: School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

Keywords

Zvonkin’s transformEuler–Maruyama approximationlow regular driftKrylov’s estimate

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 34K26: Singular perturbations 39B72: Systems of functional equations and inequalities
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 447 - 470
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bao, J., Huang, X. and Yuan, C. (2019). Convergence rate of Euler–Maruyama scheme for SDEs with Hölder–Dini continuous drifts. J. Theoret. Prob. 32, 848–871.CrossRefGoogle Scholar
Dareiotis, K., Kumar, C. and Sabanis, S. (2016). On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations. SIAM J. Numer. Anal. 54, 1840–1872.10.1137/151004872CrossRefGoogle Scholar
Flandoli, M., Gubinelli, M. and Priola, E. (2010). Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bull. Sci. Math. 134, 405–422.10.1016/j.bulsci.2010.02.003CrossRefGoogle Scholar
Gyöngy, I. and Martinez, T. (2001). On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51, 763–783.CrossRefGoogle Scholar
Gyöngy, I. and Rásonyi, M. (2011). A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stoch. Process. Appl. 121, 2189–2200.CrossRefGoogle Scholar
Halidias, N. and Kloeden, P. E. (2008). A note on the Euler–Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT 48, 51–59.CrossRefGoogle Scholar
Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063.CrossRefGoogle Scholar
Higham, D. J., Mao, X. and Yuan, C. (2007). Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45, 592–609.CrossRefGoogle Scholar
Huang, X. and Wang, F.-Y. (2019). Distribution dependent SDEs with singular coefficients. Stoch. Process. Appl. 129, 4747–4770.10.1016/j.spa.2018.12.012CrossRefGoogle Scholar
Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. London A 467, 1563–1576.CrossRefGoogle Scholar
Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Prob. 22, 1611–1641.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.CrossRefGoogle Scholar
Konakov, V. and Mammen, E. (2000). Local limits theorems for transition densities of Markov chains converging to diffusions. Prob. Theory Relat. Fields 117, 551–587.10.1007/PL00008735CrossRefGoogle Scholar
Krylov, N. V. and Röckner, M. (2005) Strong solutions of stochastic equations with singular time dependent drift. Prob. Theory Relat. Fields 131, 154–196.10.1007/s00440-004-0361-zCrossRefGoogle Scholar
Lemaire, V. and Menozzi, S. (2010). On some non asymptotic bounds for the Euler scheme. Electron. J. Prob. 15, 1645–1681.Google Scholar
Leobacher, G. and Szölgyenyi, M. (2017). A strong order 1/2 method for multidimensional SDEs with discontinuous drift. Ann. Appl. Prob. 27, 2383–2418.10.1214/16-AAP1262CrossRefGoogle Scholar
Leobacher, G. and Szölgyenyi, M. (2018). Convergence of the Euler–Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient. Numer. Math. 138, 219–239.CrossRefGoogle ScholarPubMed
Mao, X. (2015). The truncated Euler–Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 290, 370–384.CrossRefGoogle Scholar
Mao, X. and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. Imperial College Press, London.10.1142/p473CrossRefGoogle Scholar
Müller-Gronbach, T. and Yaroslavtseva, L. (2020). On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. H. Poincaré Prob. Statist. 56, 1162–1178.CrossRefGoogle Scholar
Neuenkirch, A., Szölgyenyi, M. and Szpruch, L. (2019). An adaptive Euler–Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis. SIAM J. Numer. Anal. 57, 378–403.10.1137/18M1170017CrossRefGoogle Scholar
Ngo, H.-L. and Taguchi, D. (2016). Strong rate of convergence for the Euler–Maruyama approximation of stochastic differential equations with irregular coefficients. Math. Comp. 85, 1793–1819.CrossRefGoogle Scholar
Ngo, H.-L. and Taguchi, D. (2017). On the Euler–Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. IMA J. Numer. Anal. 37, 1864–1883.Google Scholar
Pamen, O. M., Taguchi, D. (2017). Strong rate of convergence for the Euler–Maruyama approximation of SDEs with Hölder continuous drift coefficient. Stoch. Process. Appl. 127, 2542–2559.10.1016/j.spa.2016.11.008CrossRefGoogle Scholar
Röckner, M. and Zhang, X. Well-posedness of distribution dependent SDEs with singular drifts. Bernoulli 27, 1131–1158.Google Scholar
Shao, J. Weak convergence of Euler–Maruyama’s approximation for SDEs under integrability condition. Available at arXiv:1808.07250.Google Scholar
Shigekawa, I. (2004). Stochastic Analysis (Translations of Mathematical Monographs 224, Iwanami Series in Modern Mathematics). American Mathematical Society, Providence, RI.Google Scholar
Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton.Google Scholar
Suo, Y., Yuan, C. and Zhang, S.-Q. (2021). Weak convergence of Euler scheme for SDEs with low regular drift. To appear in Numer. Algor. Available at https://doi.org/10.1007/s11075-021-01206-6.CrossRefGoogle Scholar
Xie, L. and Zhang, X. (2020). Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincaré Prob. Statist. 56, 175–229.CrossRefGoogle Scholar
Yan, L. (2002). The Euler scheme with irregular coefficients. Ann. Prob. 30, 1172–1194.CrossRefGoogle Scholar
Zhang, X. (2011). Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Prob. 16, 1096–1116.CrossRefGoogle Scholar
Zhang, X. (2013). Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Rev. Mat. Iberoam. 29, 25–52.10.4171/RMI/711CrossRefGoogle Scholar
Zvonkin, A. K. (1974). A transformation of the phase space of a diffusion process that removes the drift. Math. Sb. 93, 129–149.CrossRefGoogle Scholar