Advanced Methods in Mathematical Finance

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

28 août 2018, 17:10

30m

Angers - France

Orateur

Prof. Alexander Gushchin (Steklov Mathematical Institute)

Description

Let ${\bar{N}}_t=\underset{s\le t}{sup}{N}_s$ be a running maximum of a local martingale $N$. We assume that $N$ is max-continuous, i.e. $\bar{N}$ 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:
${\sigma }_t:=inf\{s:{\bar{N}}_s>t\}.$
Then the time-changed process $M:=N\circ \sigma$ has a simple structure:
$M_t=N_{{\sigma }_t}=t\wedge W-V{1}_{\{t\ge W\}},$
where $W:={\bar{N}}_{\mathrm{\infty }}$ and $V:={\bar{N}}_{\mathrm{\infty }}-N_{\mathrm{\infty }}$ ($V$ is correctly defined on the set $\{{\bar{N}}_{\mathrm{\infty }}<\mathrm{\infty }\}$). Besides, $M_{\mathrm{\infty }}=N_{\mathrm{\infty }}$ and ${\bar{M}}_{\mathrm{\infty }}={\bar{N}}_{\mathrm{\infty }}$. 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 $(N_{\mathrm{\infty }},{\bar{N}}_{\mathrm{\infty }})$ for different subclases of max-continuous local martingales $N$, in particular, for $N$ corresponding to the Skorokhod embedding problem.