Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A weak version of Douglas Theorem with applications to finance

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