From the Skorokhod embedding problem to the Monroe theorems: New settings and solutions

The Skorokhod embedding problem was first formulated and solved by A. V. Skorokhod in 1961. Given a probability measure $\mu$ with zero mean and a finite second moment, one looks for a Brownian motion $B$ and an integrable stopping time $\tau$ such that the distribution of $B_{\tau}$ is $\mu$. During last 50 years there were found more than 20 different solutions to this (or a slightly modified) problem.
The Skorokhod embedding problem deals with embeddings a given distribution into a given process, by means of a stopping time. We are mainly interested in embeddings a whole stochastic process into, say, a Brownian motion, by means of a change of time. The Monroe theorem (1978) states that all semimartingales (and only they) are time-changed Brownian motions.
We prove a counterpart of this result for embeddings into a geometric Brownian motion. We also provide a link between this result and another Monroe’s theorem (1972) which says that a martingale can be obtained from a Brownian motion by a time change consisting of so-called minimal stopping times. Our theorem implies the same statement for all supermartingales that are bounded from below.
Our final goal is to describe all integrable processes that can be obtained from a Brownian motion by a time change consisting of minimal stopping times.

The talk is based on a joint work with Mikhail Urusov (University of Duisburg-Essen).