[Paper Review] Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs

Paul Gassiat, Harald Oberhauser+1 more|arXiv (Cornell University)|Jan 16, 2013

Stochastic processes and financial applications38 references4 citations

TL;DR

This paper establishes a viscosity PDE framework for the Skorokhod embedding problem, proving existence and minimality of Root's barrier solutions via parabolic comparison principles. It provides a self-contained, constructive PDE approach that generalizes to degenerate and time-dependent diffusions, offering new proofs and insights into reflected FBSDEs and reversed Root barriers.

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.

Motivation & Objective

• To provide a self-contained, PDE-based characterization of Root's and reversed Root barriers using viscosity solutions.
• To establish existence and minimality of Root solutions for time-inhomogeneous and degenerate diffusions, where classical methods fail.
• To generalize Rost's minimizing property (1) to non-Markovian and time-dependent settings through viscosity theory.
• To connect Skorokhod embeddings to reflected forward-backward SDEs via PDE methods.
• To offer a constructive alternative to potential-theoretic proofs of reversed Root barriers.

Proposed method

• Formulates the Skorokhod embedding problem as a nonlinear obstacle PDE with a viscosity sub- and supersolution framework.
• Applies a parabolic comparison principle to prove uniqueness and minimality of Root barriers.
• Introduces a generalized notion of regular Root barriers to ensure one-to-one correspondence with viscosity solutions.
• Uses viscosity jet characterization (Lemma 4) to handle non-smooth solutions arising from discontinuous or non-differentiable barriers.
• Leverages reflected FBSDEs to provide a probabilistic interpretation and Monte Carlo-computable framework for barrier approximation.
• Applies variational and PDE existence techniques (e.g., from Bensoussan-Lions) to construct solutions without relying on potential theory.

Experimental results

Research questions

• RQ1 Can the existence and minimality of Root’s solution to the Skorokhod embedding problem be proven via viscosity PDE methods without relying on potential theory?
• RQ2 How can the dynamics of general Skorokhod embeddings be characterized as supersolutions of nonlinear PDEs?
• RQ3 Can the reversed Root barrier be constructed constructively using PDE techniques, avoiding the classical 'filling scheme'?
• RQ4 What is the role of the parabolic comparison principle in establishing uniqueness and minimality in the viscosity setting?
• RQ5 To what extent can the PDE approach be extended to time-inhomogeneous and degenerate diffusions?

Key findings

• The paper proves that any solution to the Skorokhod embedding problem is a viscosity supersolution of a specific obstacle PDE, generalizing Rost’s excessive function approach.
• A one-to-one correspondence is established between regular Root barriers and viscosity solutions, enabling existence proofs via PDE existence theorems.
• The minimizing property (1) of Root’s solution is shown to follow directly from the parabolic comparison principle, providing a new, clean proof.
• The approach extends to time-inhomogeneous and degenerate diffusions, where classical existence and minimality results are not available.
• The reversed Root barrier is characterized as a viscosity solution, offering a constructive alternative to Rost’s potential-theoretic filling scheme.
• Numerical schemes based on the PDE formulation are implemented, and the connection to reflected FBSDEs enables Monte Carlo-based computation of barriers.