Совместные распределения обобщенных интегрируемых возрастающих процессов и их обобщенных компенсаторов

Теория вероятностей и ее применения. 2024. Т. 69. № 1. С. 3–32.

Borzykh D.

We consider the set $\Lambda$ of all edge joint distributions $\Law ([X_a, A_a], [X_b, A_b])$ at the moments $t =a$ and $t = b$ of integrable increasing processes $(X_t)_{t\in [a; b]}$ and their compensators $(A_t)_{t\in [a; b]}$, which start from an arbitrary integrable initial condition $[X_a, A_a]$.
The convexity and closure of the set $\Lambda$ in $\psi$-weak topology with a gauge function $\psi$ of linear growth are established.
Necessary and sufficient conditions are obtained that a given probability measure $\lambda$ on $\mathcal{B}(\mathbb{R}^2\times\mathbb{R}^2)$ belongs to the class of measures $\Lambda$.
The main result of the work is the following: for two measures $\mu_a$ and $\mu_b$ given on $\mathcal{B}(\mathbb{R}^2)$ necessary and sufficient conditions are obtained that the set $\Lambda$ contains the measure $\lambda$, for which $\mu_a$ and $\mu_b$ are marginal distributions.

Keywords: компенсатор, compensator, возрастающий процесс, increasing process, terminal joint distribution, Doob--Meyer decomposition, совместное терминальное распределение, разложение Дуба--Мейера, Strassen's theorem, теорема Штрассена

Publication based on the results of: Probabilistic-statistical analysis and modeling of processes with memory, Gaussian chaos, information measures (2024)