Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

A weak version of Douglas Theorem with applications to finance


Code(s) de Classification MSC:

Résumé: In this paper, we obtain a version of the Douglas Theorem for a dual system $% \left\langle X,Y\right\rangle $ of locally convex topological real vector spaces equipped with the weak topology $\sigma \left( X,Y\right) $, and we apply it to the space $L^{\infty }$ with the topology $\sigma \left( L^{\infty },L^{p}\right) $ for $p\geq 1$. Thanks to these results, we give some application to finance: we obtain a condition equivalent to the market completeness and based on the notion of extremality of measures, which permit us to give new proofs of the B\"{a}ttig-Jarrow-Jin-Madan second fundamental theorems of asset pricing. Finally, we discuss also the completeness of a slight generalisation of the Artzner and Heath example.

Mots Clés: dual systems ; weak topologies ; extremality of measures ; martingales w.r.t. a signed measure ; market completeness.

Date: 2001-05-10

Prépublication numéro: PMA-655