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

Part of: Mathematical finance Stochastic processes

Published online by Cambridge University Press:  07 August 2019

Luciano Campi  and
Claude Martini
Luciano Campi*
Affiliation:
London School of Economics and Political Science
Claude Martini*
Affiliation:
Zeliade Systems
*
*Postal address: London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK.
**Postal address: Zeliade Systems, 56 Rue Jean-Jacques Rousseau, 75001 Paris, France.
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Abstract

We investigate the supports of extremal martingale measures with prespecified 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 semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘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.

Keywords

Model-free pricingextremal measuremartingale optimal transportweak predictable representation propertycycle

MSC classification

Primary: 60G42: Martingales with discrete parameter
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 570 - 605
Copyright
© Applied Probability Trust 2019 

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