### Abstract

We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.

Original language | English |
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Pages (from-to) | 211-259 |

Number of pages | 49 |

Journal | Probability Theory and Related Fields |

Volume | 173 |

Issue number | 1-2 |

Early online date | 10 Feb 2018 |

DOIs | |

Publication status | Published - 28 Feb 2019 |

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Probability Theory and Related Fields*,

*173*(1-2), 211-259. https://doi.org/10.1007/s00440-018-0833-1