Summary: Let \(\Lambda\) be the set of all boundary joint laws \(\operatorname{Law} ([X_a, A_a], [X_b, A_b])\) at times \(t=a\) and \(t=b\) of integrable increasing processes \((X_t)_{t \in [a, b]}\) and their compensators \((A_t)_{t \in [a, b]}\), which start at the initial time from an arbitrary integrable initial condition \([X_a, A_a]\). We show that \(\Lambda\) is convex and closed relative to the \(\psi \)-weak topology with linearly growing gauge function \(\psi \). We obtain necessary and sufficient conditions for a probability measure \(\lambda\) on \(\mathcal{B}(\mathbb{R}^2 \times \mathbb{R}^2)\) to lie in the class of measures \(\Lambda \). The main result of the paper provides, for two measures \(\mu_a\) and \(\mu_b\) on \(\mathcal{B}(\mathbb{R}^2)\), necessary and sufficient conditions for the set \(\Lambda\) to contain a measure \(\lambda\) for which \(\mu_a\) and \(\mu_b\) are marginal distributions.