  In this paper, we study the existence of random periodic solutions for
nonlinear stochastic differential equations with additive white noise. We
extend the input-to-state characteristic operator of the system to the
non-autonomous stochastic differential equation via the pull-back of the
discretised stochastic differential equation. We then use the completeness of
the measurable function space which we construct skillfully and the Banach
fixed point theorem to prove the existence of a fixed point of the gain
operator. And we prove that the image for the input-to-state characteristic
operator at this fixed point is a random periodic solution for the forward
stochastic flow generated by the non-autonomous stochastic differential
equation. Finally, we present some examples.
