Electronic Journal of Probability

Two versions of the fundamental theorem of asset pricing

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60A05: Axioms; other general questions
Secondary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 91B25: Asset pricing models 91G10: Portfolio theory

Arbitrage Convex cone Equivalent martingale measure Equivalent probability measure with given marginals Finitely additive probability Fundamental theorem of asset pricing

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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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