Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

In this paper, we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable, but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [Nochetto, Savare, and Verdi, Comm.\ Pure Appl.\ Math., 2000]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.

[1]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[2]  Xuerong Mao,et al.  Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2012, J. Comput. Appl. Math..

[3]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[4]  Xuerong Mao,et al.  Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients , 2013 .

[5]  Andreas Neuenkirch,et al.  An Adaptive Euler-Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis , 2018, SIAM J. Numer. Anal..

[6]  M. Hutzenthaler,et al.  Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients , 2012, 1203.5809.

[7]  R. Pettersson.  Projection scheme for stochastic differential equations with convex constraints , 2000 .

[8]  Viorel Barbu,et al.  Nonlinear Differential Equations of Monotone Types in Banach Spaces , 2010 .

[9]  Nguyen T. Thao,et al.  Approximating and Simulating Multivalued Stochastic Differential Equations , 2004, Monte Carlo Methods Appl..

[10]  Yaozhong Hu.  Semi-Implicit Euler-Maruyama Scheme for Stiff Stochastic Equations , 1996 .

[11]  Matthias Stephan.  Yosida approximations for multivalued stochastic differential equations on Banach spaces via a Gelfand triple , 2012 .

[12]  N. Papageorgiou,et al.  Applied Nonlinear Functional Analysis , 2018 .

[13]  Paul Krée,et al.  Diffusion equation for multivalued stochastic differential equations , 1982 .

[14]  Raphael Kruse,et al.  Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition , 2015, 1509.00609.

[15]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[16]  H. Ngo,et al.  On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients , 2015, 1509.06532.

[17]  Hua Zhang.  Strong convergence rate for multivalued stochastic differential equations via stochastic theta method , 2018 .

[18]  Hoang-Long Ngo,et al.  Approximation for non-smooth functionals of stochastic differential equations with irregular drift , 2015, 1505.03600.

[19]  Michael Scheutzow.  A Stochastic Gronwall Lemma , 2013 .

[20]  OUP accepted manuscript , 2020, IMA Journal of Numerical Analysis.

[21]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .

[22]  T. Roubíček.  Nonlinear partial differential equations with applications , 2005 .

[23]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[24]  Alain Durmus,et al.  High-dimensional Bayesian inference via the unadjusted Langevin algorithm , 2016, Bernoulli.

[25]  Gunther Leobacher,et al.  Convergence of the Euler–Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient , 2016, Numerische Mathematik.

[26]  Gunther Leobacher,et al.  Correction note: A strong order 1/2 method for multidimensional SDEs with discontinuous drift , 2015, The Annals of Applied Probability.

[27]  Ying Zhang,et al.  Higher order Langevin Monte Carlo algorithm , 2018, Electronic Journal of Statistics.

[28]  T. Lelièvre,et al.  Free Energy Computations: A Mathematical Perspective , 2010 .

[29]  M. Röckner,et al.  Stochastic Partial Differential Equations: An Introduction , 2015 .

[30]  Larisa Yaroslavtseva,et al.  On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient , 2018 .

[31]  Dean S. Clark,et al.  Short proof of a discrete gronwall inequality , 1987, Discret. Appl. Math..

[32]  H. Gajewski,et al.  Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen , 1974 .

[33]  É. Moulines,et al.  The tamed unadjusted Langevin algorithm , 2017, Stochastic Processes and their Applications.

[34]  P. Kloeden,et al.  Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[35]  Gabriel J. Lord,et al.  Adaptive timestepping strategies for nonlinear stochastic systems , 2016, 1610.04003.

[36]  Andrew M. Stuart,et al.  Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations , 2002, SIAM J. Numer. Anal..

[37]  B. Leimkuhler,et al.  The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics , 2013, 1308.5814.

[38]  Benjamin Gess,et al.  Multi-valued, singular stochastic evolution inclusions , 2011, 1112.5672.

[39]  S. Sabanis.  Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients , 2013, 1308.1796.

[40]  Giuseppe Savare',et al.  A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations † , 2000 .

[41]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[42]  Jing Wu,et al.  Penalization schemes for multi-valued stochastic differential equations , 2013 .

[43]  Wolf-Jürgen Beyn,et al.  Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes , 2014, J. Sci. Comput..

[44]  Longjie Xie,et al.  Ergodicity of stochastic differential equations with jumps and singular coefficients , 2017, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[45]  Dominic Breit,et al.  Space-time approximation of stochastic $p$-Laplace systems , 2019, 1904.03134.

[46]  Konstantinos Dareiotis,et al.  On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift , 2018, 1812.04583.

[47]  F. Bernardin.  Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme , 2003 .

[48]  Roger Pettersson.  Yosida approximations for multivalued stochastic differential equations , 1995 .

[49]  Xuerong Mao,et al.  Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations , 2016, J. Comput. Appl. Math..