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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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    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

    Note: received: January 1996; final version received: April 1996
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    Keywords: Semimartingales; stochastic integrals; reverse Hölder inequalities;

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    Cited by:
    1. Jianming Xia, 2006. "Mean-variance Hedging in the Discontinuous Case," Papers math/0607775,
    2. M. Mania & R. Tevzadze & T. Toronjadze, 2007. "Mean-variance Hedging Under Partial Information," Papers math/0703424,
    3. Christoph Czichowsky, 2013. "Time-consistent mean-variance portfolio selection in discrete and continuous time," Finance and Stochastics, Springer, Springer, vol. 17(2), pages 227-271, April.
    4. M. Mania & R. Tevzadze & T. Toronjadze, 2007. "$L^2$-approximating pricing under restricted information," Papers 0708.4095,
    5. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953,
    6. Choulli, Tahir & Vandaele, Nele & Vanmaele, Michèle, 2010. "The Föllmer-Schweizer decomposition: Comparison and description," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 120(6), pages 853-872, June.
    7. Erhan Bayraktar & Ross Kravitz, 2011. "Stability of exponential utility maximization with respect to market perturbations," Papers 1107.2716,, revised Dec 2012.
    8. Arai, Takuji, 2005. "Some properties of the variance-optimal martingale measure for discontinuous semimartingales," Statistics & Probability Letters, Elsevier, Elsevier, vol. 74(2), pages 163-170, September.
    9. Vicky Henderson, 2002. "Analytical Comparisons of Option prices in Stochastic Volatility Models," OFRC Working Papers Series, Oxford Financial Research Centre 2002mf03, Oxford Financial Research Centre.
    10. Vicky Henderson & David Hobson & Sam Howison & Tino Kluge, 2003. "A Comparison of q-optimal Option Prices in a Stochastic Volatility Model with Correlation," OFRC Working Papers Series, Oxford Financial Research Centre 2003mf02, Oxford Financial Research Centre.
    11. 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.
    12. 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, Center of Finance and Econometrics, University of Konstanz 00-26, Center of Finance and Econometrics, University of Konstanz.
    13. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 296, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Schweizer, Martin, 2001. "From actuarial to financial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 31-47, February.


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