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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. 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 a description of extreme points (in a certain sense) of this set of two-dimensional laws. We also provide a link between our results and Rogers' characterization of possible joint laws of a martingale and its maximum.