  This paper concerns the stability of analytical and numerical solutions of
nonlinear stochastic delay differential equations (SDDEs). We derive sufficient
conditions for the stability, contractivity and asymptotic contractivity in
mean square of the solutions for nonlinear SDDEs. The results provide a unified
theoretical treatment for SDDEs with constant delay and variable delay
(including bounded and unbounded variable delays). Then the stability,
contractivity and asymptotic contractivity in mean square are investigated for
the backward Euler method. It is shown that the backward Euler method preserves
the properties of the underlying SDDEs. The main results obtained in this work
are different from those of Razumikhin-type theorems. Indeed, our results hold
without the necessity of constructing of finding an appropriate Lyapunov
functional.
