Concrete representation of atomic $(F_4)$ filtrations

TL;DR

This paper demonstrates that for martingales with biparameter atomic filtrations satisfying the (F_4) condition, one can construct an equivalent martingale with respect to the canonical (F_4) filtration, improving previous results by Montgomery-Smith.

Contribution

It provides a construction of a martingale with the same distribution under the canonical (F_4) filtration, independent of the underlying sequence, for biparameter atomic filtrations.

Findings

Equivalent martingales can be constructed with canonical (F_4) filtration.

The construction is a morphism of filtrations, not sequence-dependent.

Results improve upon Montgomery-Smith's theorem even in one-parameter cases.

Abstract

We prove that for any martingale with respect to a biparameter atomic filtration satisfying $(F_4)$ condition there is a martingale having the same joint distribution but with respect to the canonical $(F_4)$ filtration. Even in one parameter case our result is an improvement of the theorem due to Montgomery-Smith, since the construction gives a morphism of filtrations and does not depend on underlying sequence.