Weighted norm inequalities and hedging in incomplete markets

Martin Schweizer, Christophe Stricker, Freddy Delbaen, Pascale Monat and Walter Schachermayer
Additional contact information
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

Finance and Stochastics, 1997, vol. 1, issue 3, 227 pages

Abstract: 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.

Keywords: Semimartingales; stochastic integrals; reverse Hölder inequalities (search for similar items in EconPapers)
JEL-codes: G10 G13 (search for similar items in EconPapers)
Date: 1997
Note: received: January 1996; final version received: April 1996
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Citations: View citations in EconPapers (40)

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