On the support of extremal martingale measures with given marginals: the countable case

TL;DR

This paper characterizes extremal martingale measures with fixed marginals in a two-period setting, establishing conditions for extremality based on support structure, combinatorial properties, and the weak exact predictable representation property.

Contribution

It provides a comprehensive characterization of extremal measures in the countable support case, introducing new combinatorial conditions and linking extremality to support structure and cycles.

Findings

Equivalence between extremality and denseness in $L^1(Q)$ of certain trading strategies.

Sufficient conditions ('2-link property' and 'full erasability') for extremality with countable supports.

Necessary and sufficient conditions for the weak exact predictable representation property when the first marginal is finite.

Abstract

We investigate the supports of extremal martingale measures with pre-specified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure $Q$ and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semi-static trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (henceforth, WEP) and provide two combinatorial sufficient conditions, called "2-link property" and "full erasability", on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of $2$-net and deadlock. Finally, we study the relation between cycles and extremality.