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

TL;DR

This paper characterizes the Root solution to the multi-marginal Skorokhod embedding problem using an optimal stopping and time-reversal approach, extending the classical solution to multiple marginals with a recursive barrier construction.

Contribution

It introduces a probabilistic, time-reversal based method to solve the multi-marginal SEP, providing a complete solution with a recursive barrier characterization.

Findings

Complete solution to the n-marginal SEP via barrier hitting times

Recursive sequence of optimal stopping problems for barrier characterization

Global optimality property extended from the one-marginal case

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.