Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs

We revisit work of Rost (1976), Dupire (2005) and Cox–Wang (2013) on connections between Root’s solution of the Skorokhod embedding problem and obstacle problems. We develop an approach based on viscosity sub- and supersolutions and an accompanying comparison principle. This gives new, constructive and simple proofs of the existence and minimality properties of Root type solutions as well as their complete characterization. The approach is self-contained and covers martingale diffusions with degenerate elliptic or time-dependent volatility as well as Rost’s reversed Root barriers; it also provides insights about the dynamics of general Skorokhod embeddings by identifying them as supersolutions of certain nonlinear PDEs.