  In this paper, we study the existence of random periodic solutions for
semilinear stochastic differential equations. We identify them as solutions of
coupled forward-backward infinite horizon stochastic integral equations
(IHSIEs), using the "substitution theorem" of stochastic differential equations
with anticipating initial conditions. In general, random periodic solutions and
the solutions of IHSIEs, are anticipating. For the linear noise case, with the
help of the exponential dichotomy given in the multiplicative ergodic theorem,
we can identify them as the solutions of infinite horizon random integral
equations (IHSIEs). We then solve a localised forward-backward IHRIE in
$C(\mathbb{R}, L^2_{loc}(\Omega))$ using an argument of truncations, the
Malliavin calculus, the relative compactness of Wiener-Sobolev spaces in $C([0,
T], L^2(\Omega))$ and Schauder's fixed point theorem. We finally measurably
glue the local solutions together to obtain a global solution in $C(\mathbb{R},
L^2(\Omega))$. Thus we obtain the existence of a random periodic solution and a
periodic measure.
