F. Delbaen, W. Schachermayer
Mathematische Annalen, Vol. 312 (1998), pp. 215-250.
The Fundamental Theorem of Asset Pricing states - roughly speaking - that the absence of arbitrage possibilities for a stochastic process $S$ is equivalent to the existence of an equivalent martingale measure for $S$. It turnsout that it is quite hard to give precise and sharp versions of this theorem in proper generality, if one insists on modifying the concept of ``no arbitrage" as little as possible. It was shown in [DS94] that for a locally bounded $\R^d$-valued semi-martingale $S$ the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process $S$. It was asked whether the local boundedness assumption on $S$ may be dropped.
In the present paper we show that if we drop in this theorem the local boundedness assumption on $S$ the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma-martingale measure. The concept of sigma-martingales was introduced by Chou and Emery --- under the name of ``semimartingales de la classe $(\Sigma _m)$".
We provide an example which shows that for the validity of the theorem
in the non locally bounded case it is indeed necessary to pass to the
concept of sigma-martingales. On the other hand, we also observe that for
the applications in Mathematical Finance the notion of sigma-martingales
provides a natural framework when working with non locally bounded
The duality results which we obtained earlier are also extended to the non locally bounded case. As an application we characterize the hedgeable elements.
[DS94] --- F. Delbaen, W. Schachermayer: A General Version of the Fundamental Theorem of Asset Pricing. Math. Annalen 300 (1994): pages 463 - 520.
Sigma martingale, Arbitrage, fundamental theorem of asset pricing, Free Lunch
Primary 60G44; Secondary 46N30,46E30,90A09, 60H05
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