Almost sure characterization of martingales

Let I⊆R⁺∪{0} be an arbitrary set with 0∈I; Ξ≡(Ω,F,(F_{t})_{t∈I},P) be a complete filtered probability space such that F₀ is trivial and complete; χ be a countable family of real adapted stochastic processes on Ξ. We provide a necessary and sufficient condition for the existence of a probability measure Q, equivalent to the original measure P, under which every process X∈χ is a martingale. Furthermore, this condition is invariant under substitution of P with an equivalent probability measure; hence the theorem characterizes those real adapted stochastic processes on Ξ which can become martingales under some equivalent probability measure. The theorem we present allows us also to give a satisfactory solution to the so called Fundamental Problem of Asset Pricing which arises in Mathematical Finance; we further provide the financial interpretation of the "no-free-lunch" condition which is equivalent to the existence of a "risk-neutral measure".