Finitely Additive Supermartingales

Finitely Additive Supermartingales

The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Doléans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenou...

Finitely Additive Supermartingales Are Differences of Martingales

It is shown that any nonnegative bounded supermartingale admits a Doob-Meyer decomposition as a difference of a martingale and an adapted increasing process upon appropriate choice of a reference probability measure which may be only finitely additive. Introduction. In [Armstrong, 1983] it is shown that every bounded finitely additive supermartingale is a decreasing process with respect to some...

Finitely Additive Equivalent

Let L be a linear space of real bounded random variables on the probability space (Ω,A, P0). There is a finitely additive probability P on A, such that P ∼ P0 and EP (X) = 0 for all X ∈ L, if and only if c EQ(X) ≤ ess sup(−X), X ∈ L, for some constant c > 0 and (countably additive) probability Q on A such that Q ∼ P0. A necessary condition for such a P to exist is L− L∞ ∩ L∞ = {0}, where the cl...

On finitely additive vector measures.

In this paper we extend the well-known Vitali-Hahn-Saks and Nikodým theorems for measures to finitely additive vector-valued set functions.

Representing Finitely Additive Invariant Probabilities