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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H30 Applications of stochastic analysis (to PDE, etc.)
- 90A09 Finance, portfolios, investment
- 90A12 Price theory and market structure

**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*