Title: The fundamental theorem of asset pricing for unbounded stochastic processes
Abstract: The Fundamental Theorem of Asset Pricing states -roughly speakingthat the absence of arbitrage possibilities for a stochastic process S is equivalent t o the existence of an equivalent martingale measure for S. It turns out that it is quite hard to give precise and sharp versions of this theorem in proper generality, i f o n e 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 t h a t i f w e drop in this theorem the local boundedness assumption on S the theorem remains true if we replace the term equivalent local martingale measure by t h e t e r m equivalent sigma-martingale measure.The concept of sigma-martingales was introduced by Chou and Emery | under the name of \semimartingales de la classe ( m)".We p r o vide an example which s h o ws that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigmamartingales.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 w e obtained earlier are also extended to the non locally bounded case.As an application we c haracterize the hedgeable elements.