The Skorokhod embedding problem for inhomogeneous diffusions

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form dA(t) = mu(t, A(t)) dt + sigma (t, A(t)) dW(t). We provide sufficient conditions guaranteeing that for a given probability measure nu on 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) similar to 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.