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

Let $X$, $X_0=0$, be a nonnegative submartingale of class (D) with the Doob–Meyer decomposition $X=M+A$, where $M$ is a uniformly integrable martingale and $A$ is an integrable predictable increasing process (the compensator of $X$). We provide a characterization of possible joint laws of the terminal values $(X_\infty,A_\infty)$. It turns out that we obtain the same set of possible joint laws if we assume, in addition, that $X$ is an increasing process, or the square of a martingale. A special attention is given to extreme points (in a sense) of this set of two-dimensional laws and to a description of processes corresponding to these extreme laws. We also provide a link between our results and Rogers' characterization of possible joint laws of a martingale and its maximum.