## Electronic Journal of Probability

### 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.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 34, 21 pp.

Dates
Accepted: 29 March 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067140

Digital Object Identifier
doi:10.1214/EJP.v20-3321

Mathematical Reviews number (MathSciNet)
MR3335825

Zentralblatt MATH identifier
1326.60007

Rights

#### Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Two versions of the fundamental theorem of asset pricing. Electron. J. Probab. 20 (2015), paper no. 34, 21 pp. doi:10.1214/EJP.v20-3321. https://projecteuclid.org/euclid.ejp/1465067140

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