**Walter Schachermayer**

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## The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes.

**F. Delbaen, W. Schachermayer**

Mathematische Annalen, Vol. 312 (1998), pp. 215-250.

### Abstract:

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
processes $S$.

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.

#### Keywords:

Sigma martingale, Arbitrage, fundamental theorem of asset pricing, Free
Lunch

#### 1991 Mathematics Subject Classification:

`Primary 60G44; Secondary 46N30,46E30,90A09, 60H05`

### Preprints:

[PostScript (372 k)] [PS.gz (139 k)] [PDF (415 k)] [DOI: 10.1007/s002080050220]

Publications marked with have appeared in refereed journals.

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