# Weighted norm inequalities and hedging in incomplete markets

## Author

Listed:
• 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)

## 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.

## Suggested Citation

• Martin Schweizer & Christophe Stricker & Freddy Delbaen & Pascale Monat & Walter Schachermayer, 1997. "Weighted norm inequalities and hedging in incomplete markets," Finance and Stochastics, Springer, vol. 1(3), pages 181-227.
• Handle: RePEc:spr:finsto:v:1:y:1997:i:3:p:181-227
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## References listed on IDEAS

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1. Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187.
2. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
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6. Frey, Rüdiger, 1997. "Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility," Discussion Paper Serie B 401, University of Bonn, Germany.
7. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
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## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Michael Mania & Revaz Tevzadze, 2008. "Backward Stochastic PDEs Related to the Utility Maximization Problem," ICER Working Papers - Applied Mathematics Series 07-2008, ICER - International Centre for Economic Research.
2. Leitner Johannes, 2007. "Pricing and hedging with globally and instantaneously vanishing risk," Statistics & Risk Modeling, De Gruyter, vol. 25(4/2007), pages 1-22, October.
3. Goutte, Stéphane & Ngoupeyou, Armand, 2015. "The use of BSDEs to characterize the mean–variance hedging problem and the variance optimal martingale measure for defaultable claims," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1323-1351.
4. 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.
5. M. Mania & R. Tevzadze & T. Toronjadze, 2007. "$L^2$-approximating pricing under restricted information," Papers 0708.4095, arXiv.org.
6. David Hobson, 2004. "STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE "q"-OPTIMAL MEASURE," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 537-556.
7. 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.
8. Leitner Johannes, 2005. "Optimal portfolios with expected loss constraints and shortfall risk optimal martingale measures," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 49-66, January.
9. Vicky Henderson, 2002. "Analytical Comparisons of Option prices in Stochastic Volatility Models," OFRC Working Papers Series 2002mf03, Oxford Financial Research Centre.
10. 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.
11. 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.
12. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2.
13. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953, arXiv.org.
14. Kramkov, Dmitry & Weston, Kim, 2016. "Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2615-2633.
15. M. Mania & R. Tevzadze & T. Toronjadze, 2007. "Mean-variance Hedging Under Partial Information," Papers math/0703424, arXiv.org.
16. Jianming Xia, 2006. "Mean-variance Hedging in the Discontinuous Case," Papers math/0607775, arXiv.org.
17. 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.
18. 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 2003mf02, Oxford Financial Research Centre.
19. 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.
20. Schweizer, Martin, 2001. "From actuarial to financial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 31-47, February.

### Keywords

Semimartingales; stochastic integrals; reverse Hölder inequalities;

### JEL classification:

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

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