Higham, Desmond J.; Mao, Xuerong; Yuan, Chenggui Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. (English) Zbl 1144.65005 SIAM J. Numer. Anal. 45, No. 2, 592-609 (2007). The authors study the ability of numerical methods for stochastic differential equations to reproduce almost sure and small-moment stability. They find conditions under which the Euler-Maruyama method preserves stability properties for small timesteps. They investigate the backward Euler method and the stochastic theta method as well. Reviewer: Grigori N. Milstein (Ekaterinburg) Cited in 2 ReviewsCited in 108 Documents MSC: 65C30 Numerical solutions to stochastic differential and integral equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations Keywords:one-sided Lipschitz condition; linear growth condition; Lyapunov exponent; stochastic theta method; stochastic differential equations; Euler-Maruyama method; stability; backward Euler method PDFBibTeX XMLCite \textit{D. J. Higham} et al., SIAM J. Numer. Anal. 45, No. 2, 592--609 (2007; Zbl 1144.65005) Full Text: DOI Link