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Optional decomposition and Lagrange multipliers

  • 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)

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.

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Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 2 (1997)
Issue (Month): 1 ()
Pages: 69-81

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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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  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.
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