Finitely Additive Supermartingales

Abstract

Abstract. 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 exogenous probability measure.

Conglomerability and Representations

“…As in (7) we can construct an increasing sequence X n n∈N in S (A (λ)) such that 0 ≤ X n ≤ X and λ-converges to X. But then,…”

Quasi-martingales with a linearly ordered index set

“…The issue of the invariance of the process structure with respect to a change of the underlying filtration was also addressed in [2].…”

Asset Pricing With No Exogenous Probability Measure

“…In the theory of finitely additive measures the decomposition of Yosida and Hewitt (1952) is probably the best known and most useful result. We shall use the following generalization proved in Cassese (2006, theorem 1):…”

“…As is well known, conditional expectation is not available with respect to finitely additive probability. The following proposition (see Cassese 2006 for a different proof) introduces a concept of conditional expectation for finitely additive probabilities which is suitable for our purposes. For ease of terminology, we will refer to such operator as conditional expectation although the law of iterated expectation may fail.…”

“…These findings allow to overcome some of the difficulties involved in finitely additive expectation and, in some sense, restore the traditional properties of financial models but in purely endogenous terms. Much of these developments are based on some new results on finitely additive probabilities obtained in Cassese (2006), a measure decomposition for filtered probability spaces, Lemma 2.1, and a notion of conditional expectation for finitely additive probabilities, Proposition 2.1.…”