  In this paper, we study the existence of random periodic solutions for
semilinear stochastic differential equations. We identify these as the
solutions of coupled forward-backward infinite horizon stochastic integral
equations in general cases. We then use the argument of the relative
compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(\Omega))$ and
generalized Schauder's fixed point theorem to prove the existence of a solution
of the coupled stochastic forward-backward infinite horizon integral equations.
The condition on $F$ is then further weakened by applying the coupling method
of forward and backward Gronwall inequalities. The results are also valid for
stationary solutions as a special case when the period $\tau$ can be an
arbitrary number.\
