Concrete representation of atomic $(F_4)$ filtrations

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.