Two versions of the fundamental theorem of asset pricing

Abstract

Let L be a convex cone of real random variables on the probability space (Ω;A; P0). The existence of a probability P on A such that P ~ P0; EP |X| < G∞ and EP (X) ≤ 0 for all X ϵ L is investigated. Two types of results are provided, according to P is finitely additive or σ-additive. The main results concern the latter case (i.e., P is a σ-additive probability). If L is a linear space then -X ϵ L whenever X ϵ L, so that EP (X) = 0 turns into EP (X) = 0. Hence, the results apply to various significant frameworks, including equivalent martingale measures, equivalent probability measures with given marginals, stationary Markov chains and conditional moments.