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

Alexander M G Cox1, Jan Obłój2, Nizar Touzi3

  • 11University of Bath, Bath, UK.

Probability Theory and Related Fields
|March 19, 2019
PubMed
Summary

This study characterizes the Root solution for the Skorokhod embedding problem (SEP) using optimal stopping. It extends the solution to multiple marginals for martingale diffusions via barrier hitting times.

Keywords:
60G4060G44

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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.