ИСТИНА

Let $X$ be a nonnegative submartingale of class $(D)$ starting from $0$ with the Doob--Meyer decomposition $X=M+A$. Our main result is a characterization of the set of all possible joint distributions $\text{Law}(X_\infty, A_\infty)$. We prove, in particular, that, for every integrable law $\mu$ on the positive half-line, there exists an integrable increasing process $B$, $B_0=0$, with the compensator $C$, such that (1) $\text{Law}(B_\infty)=\mu$; (2) for every $X$ with $\text{Law}(X_\infty) = \text{Law}(B_\infty)$, $\text{Law}(A_\infty)$ is smaller than or equal to $\text{Law}(C_\infty)$ with respect to convex order; (3) for every $X$ with $\text{Law}(A_\infty) = \text{Law}(C_\infty)$, $\text{Law}(X_\infty)$ is greater than or equal to $\text{Law}(B_\infty)$ with respect to convex order. This fact has important consequences for the theory of martingale inequalities. The extreme cases in (1) and (2) correspond to the class of ``remarkable'' nonnegative submartingales introduced by Ashkan Nikeghbali (2006).