28-31 août 2018
Angers - France
Fuseau horaire Europe/Paris

The Skorokhod embedding problem and single jump martingales: a connection via change of time

28 août 2018 à 17:10
30m
Angers - France

Angers - France

Orateur

Prof. Alexander Gushchin (Steklov Mathematical Institute)

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.

Auteur principal

Prof. Alexander Gushchin (Steklov Mathematical Institute)

Documents de présentation

Aucun document.
Your browser is out of date!

Update your browser to view this website correctly. Update my browser now

×