# Weighted norm inequalities and hedging in incomplete markets

## Author Info

• 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)

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## 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.

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## Bibliographic Info

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

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## Related research

Keywords: Semimartingales; stochastic integrals; reverse Hölder inequalities;

Find related papers by JEL classification:

• G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
• G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

## References

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## Citations

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Cited by:
1. Christoph Czichowsky, 2013. "Time-consistent mean-variance portfolio selection in discrete and continuous time," Finance and Stochastics, Springer, vol. 17(2), pages 227-271, April.
2. Bayraktar, Erhan & Kravitz, Ross, 2013. "Stability of exponential utility maximization with respect to market perturbations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1671-1690.
3. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953, arXiv.org.
4. Schweizer, Martin, 2001. "From actuarial to financial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 31-47, February.
5. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series 296, Quantitative Finance Research Centre, University of Technology, Sydney.
6. Choulli, Tahir & Vandaele, Nele & Vanmaele, Michèle, 2010. "The Föllmer-Schweizer decomposition: Comparison and description," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 853-872, June.
7. Jianming Xia, 2006. "Mean-variance Hedging in the Discontinuous Case," Papers math/0607775, arXiv.org.
8. Arai, Takuji, 2005. "Some properties of the variance-optimal martingale measure for discontinuous semimartingales," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 163-170, September.
9. Badescu, Alexandru M. & Kulperger, Reg J., 2008. "GARCH option pricing: A semiparametric approach," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 69-84, August.
10. Michael Kohlmann & Shanjian Tang, 2000. "Global Adapted Solution of One-Dimensional Backward Stochastic Riccati Equations, with Application to the Mean-Variance Hedging," CoFE Discussion Paper 00-26, Center of Finance and Econometrics, University of Konstanz.

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