Exponential mean-square stability properties of stochastic linear multistep methods

The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

[1]  Angel Tocino,et al.  Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations , 2014, J. Comput. Appl. Math..

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

[3]  Ernst Hairer,et al.  Examples of Stiff Equations , 1996 .

[4]  G. Dahlquist.  Error analysis for a class of methods for stiff non-linear initial value problems , 1976 .

[5]  Desmond J. Higham,et al.  Numerical methods for nonlinear stochastic differential equations with jumps , 2005, Numerische Mathematik.

[6]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[7]  Raffaele D'Ambrosio,et al.  Drift-preserving numerical integrators for stochastic Hamiltonian systems , 2020, Adv. Comput. Math..

[8]  Desmond J. Higham,et al.  Exponential mean square stability of numerical solutions to stochastic differential equations , 2003 .

[9]  Raffaele D'Ambrosio,et al.  Nonlinear stability issues for stochastic Runge-Kutta methods , 2021, Commun. Nonlinear Sci. Numer. Simul..

[10]  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.

[11]  Inmaculada Higueras,et al.  Strong Stability for Additive Runge-Kutta Methods , 2006, SIAM J. Numer. Anal..

[12]  J. Butcher.  Thirty years of G-stability , 2006 .

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