Orateur
Description
Let $\overline N_t = \sup_{s\leq t} N_s$ be a running maximum of a local martingale $N$. We assume that $N$ is max-continuous, i.e. $\overline N$ is continuous. The Skorokhod embedding problem corresponds to a special case where $N$ is a Brownian motion stopped at a finite stopping time $\tau$. Consider the change of time generated by the running maximum:
$
\sigma_t:=\inf\,\{s\colon \overline N_s>t\}.
$
Then the time-changed process $M:=N\circ\sigma$ has a simple structure:
$
M_t=N_{\sigma_t}= t\wedge W - V1_{\{t\geq W\}},
$
where $W:=\overline N_\infty$ and $V:=\overline N_\infty-N_\infty$ ($V$ is correctly defined on the set $\{\overline N_\infty < \infty\}$). Besides, $M_\infty=N_\infty$ and $\overline M_\infty=\overline N_\infty$. This simple observation explains how we can use single jump martingales $M$ of the above form to describe properties of $N$. For example, $N$ is a closed supermartingale if and only $M$ is a martingale and the negative part of $W-V$ is integrable. Another example shows how to connect the Dubins-Gilat construction of a martingale whose supremum is given by the Hardy-Littlewood maximal function and the Azéma-Yor construction in the Skorokhod embedding problem.
Summary
We establish a connection between the sets of possible joint distributions of pairs $(N_\infty, \overline N_\infty)$ for different subclases of max-continuous local martingales $N$, in particular, for $N$ corresponding to the Skorokhod embedding problem.
