Finitely Additive Equivalent Martingale Measures

Let L be a linear space of real bounded random variables on the probability space $(\varOmega ,\mathcal{A},P_{0})$ . There is a finitely additive probability P on $\mathcal{A}$ such that P∼P 0 and E P (X)=0 for all X∈L if and only if cE Q (X)≤ess sup (−X), X∈L, for some constant c>0 and (countably additive) probability Q on $\mathcal{A}$ such that Q∼P 0. A necessary condition for such a P to exist is $\overline{L-L_{\infty}^{+}}\cap L_{\infty}^{+}=\{0\}$ , 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≪P 0 and E P (X)=0 for all X∈L if and only if ess sup (X)≥0 for all X∈L.