Minimal Root's embeddings for general starting and target distributions

TL;DR

This paper extends the theory of Root's embeddings in Skorokhod problems by removing the uniform integrability assumption, replacing it with a minimality condition, and generalizing to multi-marginal cases.

Contribution

It introduces a minimality-based criterion for Root's solutions without convex ordering, broadening applicability to more general distributions and multi-marginal problems.

Findings

Provides necessary and sufficient conditions for minimality in Root's embeddings.

Extends results to multi-marginal embedding problems.

Discusses the optimality of minimal solutions.

Abstract

Most results regarding Skorokhod embedding problems (SEP) so far rely on the assumption that the corresponding stopped process is uniformly integrable, which is equivalent to the convex ordering condition $\mathrm{U}^{\mu}\leq\mathrm{U}^{\nu}$ when the underlying process is a local martingale. In this paper, we study the existence, construction of Root's solutions to SEP, in the absence of this convex ordering condition. We replace the uniform integrability condition by the minimality condition (Monroe,1972), as the criterion of "good" solutions. A sufficient and necessary condition (in terms of local time) for minimality is given. We also discuss the optimality of such minimal solutions. These results extend the generality of the results given by Cox and Wang [2013] and Gassiat et al. [2015]. At last, we extend all the results mentioned above to multi-marginal embedding problems based on the work of Cox et al. [2018].