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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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    Cited by:

    1. Frank Bosserhoff & Mitja Stadje, 2019. "Robustness of Delta Hedging in a Jump-Diffusion Model," Papers 1910.08946, arXiv.org, revised Apr 2022.
    2. 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.
    3. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
    4. Riedel, Frank, 2010. "Optimal Stopping under Ambiguity," Center for Mathematical Economics Working Papers 390, Center for Mathematical Economics, Bielefeld University.
    5. Sabrina Mulinacci, 2011. "The efficient hedging problem for American options," Finance and Stochastics, Springer, vol. 15(2), pages 365-397, June.
    6. Karim El Moutaouakil & Abdellatif El Ouissari & Vasile Palade & Anas Charroud & Adrian Olaru & Hicham Baïzri & Saliha Chellak & Mouna Cheggour, 2023. "Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population," Mathematics, MDPI, vol. 11(13), pages 1-28, July.
    7. 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.
    8. 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.
    9. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
    10. Bruno Bouchard & Xiaolu Tan, 2021. "A quasi-sure optional decomposition and super-hedging result on the Skorokhod space," Finance and Stochastics, Springer, vol. 25(3), pages 505-528, July.
    11. 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.
    12. 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.
    13. Jun Sekine, 2012. "Long-term optimal portfolios with floor," Finance and Stochastics, Springer, vol. 16(3), pages 369-401, July.
    14. Marco Rodrigues, 2025. "Robust Hedging of American Options via Aggregated Snell Envelopes," Papers 2506.14553, arXiv.org.
    15. Joao Amaro de Matos & Ana Lacerda, 2004. "Dry markets and superreplication bounds of American derivatives," Nova SBE Working Paper Series wp461, Universidade Nova de Lisboa, Nova School of Business and Economics.

    More about this item

    Keywords

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    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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