A note on extremality and completeness in financial markets with infinitely many risky assets

Luciano Campi (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 FiME Lab - Laboratoire de Finance des Marchés d'Energie

In this paper we study the interplay existing between completeness of financial markets with infinitely many risky assets and extremality of equivalent martingale measures. In particular, we obtain a version of the Douglas- Naimark Theorem for a dual system ‹X, Y› of locally convex topological real vector spaces equipped with the weak topology σ(X, Y), and we apply it to the space L∞ with the topology σ(L∞, Lp) for p≥1. Thanks to these results, we obtain a condition equivalent to the market completeness and based on the notion of extremality of measures, which allows us to give new and simpler proofs of the second fundamental theorems of asset pricing. Finally, we discuss also the completeness of a slight generalization of the Artzner and Heath example.

Mots clés: weak topologies; dual system; market completeness; martingale measures; extremality of measures

Domaines: Mathématiques [math]

Luciano Campi. A note on extremality and completeness in financial markets with infinitely many risky assets. Rendiconti del Seminario Matematico della Università di Padova, 2004, 112. ⟨hal-01715042⟩