Joint Distributions of Generalized Integrable Increasing Processes and Their Generalized Compensators

Abstract

Theory of Probability &Its Applications, Volume 69, Issue 1, Page 1-24, May 2024.
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 $\mathcal{B}(\mathbf{R}^2 \times \mathbf{R}^2)$ to lie in the class of measures $\Lambda$. The main result of the paper provides, for two measures $\mu_a$ and $\mu_b$ on $\mathcal{B}(\mathbf{R}^2)$, necessary and sufficient conditions for the set $\Lambda$ to contain a measure $\lambda$ for which $\mu_a$ and $\mu_b$ are marginal distributions.