The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

Area of Science:

• Probability Theory
• Stochastic Analysis
• Mathematical Finance

Background:

• The Skorokhod embedding problem (SEP) is a fundamental challenge in probability theory.
• Existing solutions, like the Root solution, primarily address the one-marginal case.
• Extending these solutions to multiple marginals remains a significant open problem.

Purpose of the Study:

• To provide a complete probabilistic characterization of the Root solution to the Skorokhod embedding problem.
• To develop a novel approach for solving the n-marginal Skorokhod embedding problem.
• To extend the global optimality property of the Root solution to the multi-marginal case.

Main Methods:

• Optimal stopping formulation
• Probabilistic methods with a tailored time-reversal argument
• Analysis of first hitting times of barrier sets by time-space processes

Main Results:

• A complete characterization of the Root solution to the Skorokhod embedding problem (SEP) using optimal stopping.
• A novel solution to the n-marginal SEP is established using barrier hitting times.
• The proposed solution demonstrates a global optimality property, extending the one-marginal Root case.

Conclusions:

• The study successfully extends the Root solution of the SEP to the n-marginal case for general one-dimensional martingale diffusions.
• The optimal stopping framework provides a powerful tool for addressing complex problems in stochastic analysis.
• This work offers a significant advancement in understanding and solving multi-dimensional marginal problems in probability theory.