The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.

Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ (with explicit SDE or transition semigroup, or infinitesimal generator), and a stopping time $\tau$ (let's say a.s. finite).

I was wondering if (or when) it was possible to find a process $Y_t$ such that $Y_1=X_{\tau}$ and where the dynamics of $Y_t$ is explicitely know (aka an exlicit SDE for $Y$, or its transition semigroup).

Best Regards