# Optional decomposition and Lagrange multipliers

## Author

Listed:
• H. Föllmer

(Institut für Mathematik, Humboldt Universität, Unter den Linden 6, D-10099 Berlin, Germany)

• Y.M. Kabanov

(Central Economics and Mathematics Institute of the Russian Academy of Sciences, Moscow)

## Abstract

Let ${\cal Q}$ be the set of equivalent martingale measures for a given process $S$, and let $X$ be a process which is a local supermartingale with respect to any measure in ${\cal Q}$. The optional decomposition theorem for $X$ states that there exists a predictable integrand $\varphi$ such that the difference $X-\varphi\cdot S$ is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.

## Suggested Citation

• H. Föllmer & Y.M. Kabanov, 1997. "Optional decomposition and Lagrange multipliers," Finance and Stochastics, Springer, vol. 2(1), pages 69-81.
• Handle: RePEc:spr:finsto:v:2:y:1997:i:1:p:69-81
Note: received: January 1996; final version received: June 1997
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## References listed on IDEAS

as
1. Ernst Eberlein & Jean Jacod, 1997. "On the range of options prices (*)," Finance and Stochastics, Springer, vol. 1(2), pages 131-140.
2. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany.
Full references (including those not matched with items on IDEAS)

## Citations

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

1. Jun Sekine, 2012. "Long-term optimal portfolios with floor," Finance and Stochastics, Springer, vol. 16(3), pages 369-401, July.
2. Alexander Chigodaev, 2016. "Recursive Method for Guaranteed Valuation of Options in Deterministic Game Theoretic Approach," HSE Working papers WP BRP 53/FE/2016, National Research University Higher School of Economics.
3. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
4. Bank, Peter & Riedel, Frank, 1999. "Optimal consumption choice under uncertainty with intertemporal substitution," SFB 373 Discussion Papers 1999,71, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
5. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
6. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
7. Filipovic, Damir & Kupper, Michael, 2007. "Monotone and cash-invariant convex functions and hulls," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 1-16, July.
8. Riedel, Frank, 2010. "Optimal Stopping under Ambiguity," Center for Mathematical Economics Working Papers 390, Center for Mathematical Economics, Bielefeld University.
9. Kohlmann, Michael & Niethammer, Christina R., 2007. "On convergence to the exponential utility problem," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1813-1834, December.
10. Sabrina Mulinacci, 2011. "The efficient hedging problem for American options," Finance and Stochastics, Springer, vol. 15(2), pages 365-397, June.
11. Matos, Joao Amaro de & Lacerda, Ana, 2004. "Dry Markets and Superreplication Bounds of American Derivatives," FEUNL Working Paper Series wp461, Universidade Nova de Lisboa, Faculdade de Economia.

### Keywords

Optional decomposition; semimartingale; equivalent martingale measure; Hellinger process; Lagrange multiplier;

### JEL classification:

• G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
• G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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