Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $$\frac{1}{2}$$12 for the two-step BDF-Maruyama scheme and for the backward Euler–Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $$\tfrac{3}{2}$$32-volatility model from finance and a two dimensional problem related to Galerkin approximation of SPDE, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.

[1]  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.

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

[3]  M. Hutzenthaler,et al.  On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients , 2014, 1401.0295.

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

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

[6]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[7]  Etienne Emmrich.  Two-step Bdf Time Discretisation of Nonlinear Evolution Problems Governed by Monotone Operators with Strongly Continuous Perturbations , 2009, Comput. Methods Appl. Math..

[8]  P. Kloeden,et al.  Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients , 2010, 1010.3756.

[9]  Zhongqiang Zhang,et al.  A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications , 2012, SIAM J. Numer. Anal..

[10]  Raphael Kruse,et al.  Characterization of bistability for stochastic multistep methods , 2012 .

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

[12]  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 .

[13]  Evelyn Buckwar,et al.  Multistep methods for SDEs and their application to problems with small noise , 2006, SIAM J. Numer. Anal..

[14]  Joanna Goard,et al.  STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS , 2013 .

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

[16]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[17]  Pierre Henry-Labordere,et al.  Solvable local and stochastic volatility models: supersymmetric methods in option pricing , 2005, cond-mat/0511028.

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

[19]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

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

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