Multilevel Monte Carlo method for ergodic SDEs without contractivity

This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon-Nykodim derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to $T.$ Moreover, the variance of the new level estimators increase linearly in $T,$ which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to $O(\varepsilon^{-2}|\log \varepsilon|^{2})$ from $O(\varepsilon^{-3}|\log \varepsilon|)$ of the standard Monte Carlo method. Numerical experiments support our analysis.

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

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

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

[4]  A. Eberle,et al.  Couplings and quantitative contraction rates for Langevin dynamics , 2017, The Annals of Probability.

[5]  M. Giles,et al.  Multilevel Path Simulation for Jump-Diffusion SDEs , 2012 .

[6]  M. V. Tretyakov,et al.  Computing ergodic limits for Langevin equations , 2007 .

[7]  P. Shahabuddin,et al.  Chapter 11 Rare-Event Simulation Techniques: An Introduction and Recent Advances , 2006, Simulation.

[8]  Mark Denny,et al.  Introduction to importance sampling in rare-event simulations , 2001 .

[9]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[10]  Jonathan C. Mattingly,et al.  Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , 2002 .

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

[12]  Niels Richard Hansen.  Geometric ergodicity of discrete-time approximations to multivariate diffusions , 2003 .

[13]  Patrik Andersson,et al.  Unbiased simulation of stochastic differential equations using parametrix expansions , 2017 .

[14]  Marc Yor,et al.  Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times , 1982 .

[15]  V. Lemaire.  An adaptive scheme for the approximation of dissipative systems , 2005, math/0502317.

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

[17]  D. Talay.  Second-order discretization schemes of stochastic differential systems for the computation of the invariant law , 1990 .

[18]  S. Meyn,et al.  Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes , 1993, Advances in Applied Probability.

[19]  Francisco Bernal,et al.  Multilevel Estimation of Expected Exit Times and Other Functionals of Stopped Diffusions , 2017, SIAM/ASA J. Uncertain. Quantification.

[20]  Riccardo Gasparotto,et al.  Optimised Importance Sampling in Multilevel Monte Carlo , 2015 .

[21]  Christian Soize,et al.  The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions , 1994, Series on Advances in Mathematics for Applied Sciences.

[22]  G. Roberts,et al.  Exact simulation of diffusions , 2005, math/0602523.

[23]  Michael B. Giles.  Multilevel Monte Carlo methods , 2015, Acta Numerica.

[24]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[25]  Andrew M. Stuart,et al.  Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations , 2009, SIAM J. Numer. Anal..

[26]  Ahmed Kebaier,et al.  Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation , 2015, Methodology and Computing in Applied Probability.

[27]  Mu-Fa Chen,et al.  Coupling Methods for Multidimensional Diffusion Processes , 1989 .

[28]  Michael B. Giles,et al.  Adaptive Euler-Maruyama Method for SDEs with Non-globally Lipschitz Drift: Part I, Finite Time Interval , 2016, 1703.06743.