Two versions of the fundamental theorem of asset pricing

Abstract

Let $L$ be a convex cone of real random variables on the probability space $(\Omega,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $\begin{equation*} P \sim P_0,\quad E_P\left|X\right| < \infty\,\textrm{and} E_P(X)\leq 0\, \mathrm{for all} X \in L \end{equation*}$ is investigated. Two results are provided. In the first, $P$ is a finitely additive probability, while $P$ is $\sigma$-additive in the second. If $L$ is a linear space then $-X\in L$ whenever $X \in L$, so that $E_P(X)\leq 0$ turns into $E_P(X)=0$. Hence, the results apply to various significant frameworks, including equivalent martingale measures and equivalent probability measures with given marginals.