  We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous
diffusion $X$: given a distribution $\rho$, we construct a stopping time $\tau$
such that the stopped process $X_{\tau}$ has the distribution $\rho$. Our
solution method makes use of martingale representations (in a similar way to
Bass (In Seminar on Probability XVII. Lecture Notes in Math. 784 (1983) 221-224
Springer) who solves the SEP for Brownian motion) and draws on law uniqueness
of weak solutions of SDEs. Then we ask if there exist solutions of the SEP
which are respectively finite almost surely, integrable or bounded, and when
does our proposed construction have these properties. We provide conditions
that guarantee existence of finite time solutions. Then, we fully characterize
the distributions that can be embedded with integrable stopping times. Finally,
we derive necessary, respectively sufficient, conditions under which there
exists a bounded embedding.
