It was very easy to show that the linear scalar stochastic differential equation (SDE)
dx(t) = b x(t) dB(t)
is almost surely exponentially stable as long as b ≠ 0.
However, it was nontrivial for Mohammed and Scheutzow (1997) to show if the corresponding linear scalar stochastic differential delay equation (SDDE)
dx(t) = b x(t -) dB(t)
is almost surely exponentially stable for sufficiently small .
There has been a very little progress in this topic since 1997. This talk will report some recent developments.