  This paper analyzes the limiting behavior of stochastic linear-quadratic
optimal control problems in finite time horizon $[0,T]$ as
$T\rightarrow\infty$. The so-called turnpike properties are established for
such problems, under stabilizability condition which is weaker than the
controllability, normally imposed in the similar problem for ordinary
differential systems. In dealing with the turnpike problem, a crucial issue is
to determine the corresponding static optimization problem. Intuitively
mimicking deterministic situations, it seems to be natural to include both the
drift and the diffusion as constraints in the static optimization problem.
However, this would lead us to a wrong direction. It is found that the correct
static problem should contain the diffusion as a part of the objective
function, which reveals a deep feature of the stochastic turnpike problem.
