We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.
"An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift." Electron. J. Probab. 20 1 - 38, 2015. https://doi.org/10.1214/EJP.v20-3758