Market free lunch and large financial markets.pdfVIP

a r X i v : m a t h / 0 7 0 2 4 0 9 v 1 [ m a t h .P R ] 1 4 F e b 2 0 0 7 The Annals of Applied Probability 2006, Vol. 16, No. 4, 2055–2077 DOI: 10.1214c? Institute of Mathematical Statistics, 2006 MARKET FREE LUNCH AND LARGE FINANCIAL MARKETS By Irene Klein University of Vienna The main result of the paper is a version of the fundamental the- orem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage are character- ized in terms of no asymptotic market free lunch; the difference lies in the set of utilities. In particular, it is shown directly that no asymp- totic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free lunch. In principle, the paper can be seen as the large financial market analogue of [Math. Finance 14 (2004) 351–357] and [Math. Finance 16 (2006) 583–588]. 1. Introduction. In a semimartingale model of a securities market, Frittelli [5] introduced the notion of no market free lunch (NMFL) which is a condition of no arbitrage type induced by the preferences of all agents in the market. Preferences are given by expected utility, where the expectation is taken with respect to an equivalent probability measure which, together with the utility function, determines an agent. Frittelli gave characteriza- tions of the well-known notions of no arbitrage (NA) and no free lunch with vanishing risk (NFLVR) in terms of NMFL, the difference between the characterizations depending on the class of utility functions. Moreover, he provided a new version of the fundamental theorem of asset pricing (FTAP), that is, he gave a direct proof of the equivalence between NMFL with respect to monotone concave utility functions and the existence of a (local/sigma) martingale measure. The proof is not based on any o