A BSDE approach to the Skorokhod embedding problem for the Brownian motion with drift

We solve Skorokhod's embedding problem for Brownian motion with linear drift (W-t + kappa t)(t >= 0) by means of techniques of stochastic control theory. The search for a stopping time T such that the law of W-T + kappa T coincides with a prescribed law mu possessing the first moment is based on solutions of backward stochastic differential equations of quadratic type. This new approach generalizes an approach by Bass [3] of the classical version of Skorokhod's embedding problem using martingale representation techniques.