The joint law of terminal values of a nonnegative submartingale and its compensator

We characterize the set $W$ of possible joint laws of terminal values of a nonnegative submartingale $X$ of class $(D)$, starting at 0, and the predictable increasing process (compensator) from its Doob–Meyer decomposition. The set of possible values remains the same under certain additional constraints on $X$, for example, under the condition that $X$ is an increasing process or a squared martingale. Special attention is paid to extremal (in a certain sense) elements of the set $W$ and to the corresponding processes. We relate also our results with Rogers's results on the characterization of possible joint values of a martingale and its maximum.