Let \$L\$ be a linear space of real bounded random variables on the probability space \$(Omega,mathcalA,P_0)\$. There is a finitely additive probability \$P\$ on \$mathcalA\$, such that \$Psim P_0\$ and \$E_P(X)=0\$ for all \$Xin L\$, if and only if \$c,E_Q(X)leq extess sup(-X)\$, \$Xin L\$, for some constant \$c&gt;0\$ and (countably additive) probability \$Q\$ on \$mathcalA\$ such that \$Qsim P_0\$. A necessary condition for such a \$P\$ to exist is \$overlineL-L_infty^+,cap L_infty^+=\$, 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 \$mathcalA\$, such that \$Pll P_0\$ and \$E_P(X)=0\$ for all \$Xin L\$, if and only if \$ extess sup(X)geq 0\$ for all \$Xin L\$.

### Finitely additive equivalent martingale measures

#### Abstract

Let \$L\$ be a linear space of real bounded random variables on the probability space \$(Omega,mathcalA,P_0)\$. There is a finitely additive probability \$P\$ on \$mathcalA\$, such that \$Psim P_0\$ and \$E_P(X)=0\$ for all \$Xin L\$, if and only if \$c,E_Q(X)leq extess sup(-X)\$, \$Xin L\$, for some constant \$c>0\$ and (countably additive) probability \$Q\$ on \$mathcalA\$ such that \$Qsim P_0\$. A necessary condition for such a \$P\$ to exist is \$overlineL-L_infty^+,cap L_infty^+=\$, 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 \$mathcalA\$, such that \$Pll P_0\$ and \$E_P(X)=0\$ for all \$Xin L\$, if and only if \$ extess sup(X)geq 0\$ for all \$Xin L\$.
##### Scheda breve Scheda completa Scheda completa (DC)
2013
Berti Patrizia; Pratelli Luca; Rigo Pietro
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/734028`
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