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Weighted norm inequalities and hedging in incomplete markets

  • Martin Schweizer

    (TU Berlin, Fachbereich Mathematik, Strasse des 17. Juni 136, D-10623 Berlin, Germany)

  • Christophe Stricker

    (Laboratoire de Mathématiques, URA CNRS 741, 16 Route de Gray, F-25030 Besançon Cedex, France)

  • Freddy Delbaen

    (Department of Mathematics, Eidgenössische Technische Hochschule Zürich, CH-8092 Zürich, Switzerland)

  • Pascale Monat

    (Laboratoire de Mathématiques, URA CNRS 741, 16 Route de Gray, F-25030 Besançon Cedex, France)

  • Walter Schachermayer

    (Universität Wien, Brünnerstrasse 72, A-1210 Wien, Austria)

Registered author(s):

    Let $X$ be an ${\Bbb R}^d$-valued special semimartingale on a probability space $(\Omega , {\cal F} , ({\cal F} _t)_{0 \leq t \leq T} ,P)$ with canonical decomposition $X=X_0+M+A$. Denote by $G_T(\Theta )$ the space of all random variables $(\theta \cdot X)_T$, where $\theta $ is a predictable $X$-integrable process such that the stochastic integral $\theta \cdot X$ is in the space ${\cal S} ^2$ of semimartingales. We investigate under which conditions on the semimartingale $X$ the space $G_T(\Theta )$ is closed in ${\cal L} ^2(\Omega , {\cal F} ,P)$, a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of $G_T(\Theta )$ in ${\cal L} ^2(P)$. Most of these conditions deal with BMO-martingales and reverse Hölder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Föllmer-Schweizer decomposition.

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    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 1 (1997)
    Issue (Month): 3 ()
    Pages: 181-227

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    Handle: RePEc:spr:finsto:v:1:y:1997:i:3:p:181-227
    Note: received: January 1996; final version received: April 1996
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