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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

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The question as it is asked is really general so do not hesitate to give particular cases where a solution is attainable. – The Bridge Mar 1 '11 at 9:48
May be this could do the (theoretical) trick, define $\mathcal{G_t}=\mathcal{F}_t\vee \tau$ the augmented filtration of the original stocahstic basis, then defining $Y_t=E[X_\tau| \mathcal{G}_t]$ would be correct answer. anyway this is still theoretical since this doesn't give analytical form to the $Y$ process – The Bridge Mar 1 '11 at 14:57

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