Different kinds of two-parameter martingales

This paper is devoted to the study of all the different classes of two-parameter martingales which were introduced during the last decade. The problem of the relations between these classes is completely solved under the F-4 assumption and counter-examples are given in order to point out the differences between these classes. Characterizations in terms of Doob-Meyer-Cairoli decompositions are obtained. In the case where the filtration is generated by a two-parameter Poisson process, we prove that the class of strong martingales coincides with the class of martingales of direction independent variation.