Finitely Additive Equivalent Martingale Measures - Patrizia Berti; Luca Pratelli; Pietro Rigo (Università degli Studi di Pavia, Dipartimento di Economia Politica e Metodi Quantitativi (EPMQ), 2010)
Let L be a linear space of real bounded random variables on the probability space (omega,A, P0). There is a finitely additive probability P on A, such that P tilde P0 and EP (X) = 0 for all X in L, if and only if cEQ(X) = ess sup(-X), X in L, for some constant c > 0 and (countably additive) probability Q on A such that Q tilde P0. A necessary condition for such a P to exist is L - L+(inf) n L+(inf) = 0, where the closure is in the norm-topology. If P0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on A, such that P << P0 and EP (X) = 0 for all X in L, if and only if ess sup(X) = 0 for all X in L.
Keywords: 330; Arbitrage; de Finettis coherence principle; equivalent martingale measure; finitely additive probability; fundamental theorem of asset pricing