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

TL;DR

This paper explores the connection between Root's solution to the Skorokhod embedding problem and obstacle PDEs, introducing a viscosity solution approach that generalizes to complex diffusions and offers new proofs of existence and minimality.

Contribution

It develops a viscosity solution framework for Root's barriers, providing a unified PDE-based method for existence, minimality, and characterization of solutions, extending to degenerate and time-dependent cases.

Findings

Complete characterization of Root barriers via viscosity solutions

New PDE proofs of existence and minimality of barriers

Insights into Skorokhod embedding dynamics

Abstract

We revisit work of Rost, Dupire and Cox--Wang 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 a complete characterization of (reversed) Root barriers and leads to new proofs of existence as well as minimality of such barrier solutions by pure PDE methods. The approach is self-contained and general enough to cover martingale diffusions with degenerate elliptic or time-dependent volatility; it also provides insights about the dynamics of general Skorokhod embeddings.