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 = μ(t,A_t)dt + σ(t,A_t)dW_t. We provide sufficient conditions guaranteeing that for a given probability measure ν on R there exists a bounded stopping time τ and a real a such that the solution (A_t) of the SDE with initial value a satisfies A_τ ∼ ν. 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.