[Paper Review] On the Root solution to the Skorokhod embedding problem given full marginals

Alexandre Richard, Xiaolu Tan+1 more|arXiv (Cornell University)|Oct 23, 2018

Nonlinear Partial Differential Equations23 references4 citations

TL;DR

This paper characterizes the Root solution to the Skorokhod embedding problem for full marginals on a compact time interval by deriving a parabolic PDE that describes the potential function of the optimal stopping time barrier. The authors establish the limit behavior of finite-marginal Root solutions and prove that the limiting stopping times satisfy a time-homogeneous hitting condition, providing a continuous, optimal embedding under convex order constraints.

ABSTRACT

This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Obl\\'oj and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

Motivation & Objective

• To extend the Root solution of the Skorokhod embedding problem from finite to full marginals on a compact time interval.
• To characterize the limiting barrier structure of the Root embedding when marginals are given at every time point in [0,1].
• To establish a connection between the limiting Root solution and a parabolic partial differential equation governing the potential function.
• To prove continuity and optimality properties of the embedding under convex order and right-continuity assumptions.
• To address the challenge of non-ordered barriers in multiple-marginal embeddings by analyzing convergence of finite-marginal solutions.

Proposed method

• Leverages limiting arguments based on the finite-marginal Root solution construction by Cox, Obłój, and Touzi (2013).
• Uses a variational inequality approach to derive the potential function of the Root barrier in the limit.
• Applies tightness and weak convergence arguments to show convergence of finite-marginal Root barriers to a limiting barrier set.
• Establishes that the limiting stopping times satisfy a time-homogeneous hitting condition: $\sigma^\infty_s = \inf\{t \geq \sigma^\infty_s : (t,W_t) \in \mathcal{R}_s\}$ a.s.
• Employs the monotonicity principle from optimal transport to justify the optimality of the limiting solution.
• Uses the fact that $s \mapsto \mu_s$ is continuous and increasing in convex order to ensure existence of a martingale embedding.

Experimental results

Research questions

• RQ1 How can the Root solution to the Skorokhod embedding problem be extended from finitely many marginals to a continuous family of full marginals on [0,1]?
• RQ2 What PDE characterizes the potential function of the limiting Root barrier in the full-marginal case?
• RQ3 Under what conditions does the sequence of finite-marginal Root barriers converge to a limiting barrier in the full-marginal setting?
• RQ4 Can the limiting embedding preserve optimality (e.g., minimal variance) in the full-marginal context?
• RQ5 Is the limiting stopping time process continuous and consistent with the hitting time of a time-space domain?

Key findings

• The limiting Root solution satisfies $\sigma^\infty_s = \inf\{t \geq \sigma^\infty_s : (t,W_t) \in \mathcal{R}_s\}$ almost surely, indicating a time-homogeneous hitting condition.
• The potential function of the limiting barrier is characterized by a parabolic PDE derived from the limiting behavior of finite-marginal solutions.
• For each fixed $s \in [0,1]$, the sequence of finite-marginal Root barriers $\mathcal{R}^n_s$ converges along a subsequence to a limit barrier $\mathcal{R}^\infty_s$.
• The limiting stopping time process $\sigma^\infty_s$ is continuous on $[0,1] \setminus N$ for some countable null set $N$, ensuring regularity of the embedding.
• The limiting embedding satisfies $\mathcal{L}(W_{\sigma^\infty_s}) = \mu_s$ for all $s \in [0,1]$, confirming correct marginal law propagation.
• The limit solution inherits the optimality of the Root solution: it minimizes variance among all embeddings satisfying the full-marginal constraints.