In this talk we will discuss the the linear stability properties of the Euler methods for stochastic differential equations(SDEs) applied with constant stepsize. We extend the T-stability defined by Saito and Mitsui from weak solutions to strong solutions. The extended definition of T-stability is equivalent to the asymptotic stability property of a numerical method applied to the linear test equation, which is also studied by Higham. As the T-stability property has a strong relationship with the number of steps, T(A)-stability is defined in this paper. We prove that the implicit Euler method is T-stable for certain values of the linear test problem and give the T(A)-stability regions of the Euler methods. The numerical results indicate that the definition of T(A)-stability is meaningful.

MINISIMPOSIUM SESSION: 7. Numerical methods for Stochastic equations