  We solve the Skorokhod embedding problem for a class of stochastic processes
satisfying an inhomogeneous stochastic differential equation (SDE) of the form
$d A_t =\mu (t, A_t) d t + \sigma(t, A_t) d W_t$. We provide sufficient
conditions guaranteeing that for a given probability measure $\nu$ on
$\mathbb{R}$ there exists a bounded stopping time $\tau$ and a real $a$ such
that the solution $(A_t)$ of the SDE with initial value $a$ satisfies $A_\tau
\sim \nu$. We hereby distinguish the cases where $(A_t)$ is a solution of the
SDE in a weak or strong sense. Our construction of embedding stopping times is
based on a solution of a fully coupled forward-backward SDE. We use the
so-called method of decoupling fields for verifying that the FBSDE has a unique
solution. Finally, we sketch an algorithm for putting our theoretical
construction into practice and illustrate it with a numerical experiment.
