Almost sure characterization of Martingales

Let be an arbitrary set with be a complete filtered probability space such that F 0 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 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”.