Finitely additive equivalent martingale measures

TL;DR

This paper characterizes when finitely additive equivalent martingale measures exist for a given linear space of bounded random variables, linking their existence to conditions involving countably additive measures and essential supremum constraints.

Contribution

It provides necessary and sufficient conditions for the existence of finitely additive equivalent martingale measures, extending classical results to the finitely additive setting.

Findings

Existence of finitely additive measures characterized by a constant and a countably additive measure.

Necessary condition involves the intersection of the closure of a certain set with positive bounded functions.

Sufficient condition holds when the underlying probability is atomic.

Abstract

Let $L$ be a linear space of real bounded random variables on the probability space $(\Omega,\mathcal{A},P_0)$. There is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\sim P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $c\,E_Q(X)\leq\text{ess sup}(-X)$, $X\in L$, for some constant $c>0$ and (countably additive) probability $Q$ on $\mathcal{A}$ such that $Q\sim P_0$. A necessary condition for such a $P$ to exist is $\bar{L-L_\infty^+}\,\cap L_\infty^+=\{0\}$, where the closure is in the norm-topology. If $P_0$ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\ll P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $\text{ess sup}(X)\geq 0$ for all $X\in L$.