Summary
LetM(X) be the family of all equivalent local martingale measuresQ for some locally boundedd-dimensional processX, andV be a positive process. The main result of the paper (Theorem 2.1) states that the processV is a supermartingale whateverQ∈M(X), if and only if this process admits the following decomposition:
whereH is an integrand forX, andC is an adapted increasing process. We call such a representationoptional because, in contrast to the Doob-Meyer decomposition, it generally exists only with an adapted (optional) processC. We apply this decomposition to the problem of hedging European and American style contingent claims in the setting ofincomplete security markets.
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This research was performed at the University of Bonn under contract with SFB 303 and is partially supported by the Russian Foundation of Fundamental Researches, Grant 93-011-1440 and by the International Science Foundation, Grant No. MMK000
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Kramkov, D.O. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Th. Rel. Fields 105, 459–479 (1996). https://doi.org/10.1007/BF01191909
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DOI: https://doi.org/10.1007/BF01191909