  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].
