## Sufficient conditions for uniform integrability of non-negative two- parametric martingales.(Russian)Zbl 0635.60048

The paper contains exponential representations for two-parameter continuous martingales $$x_ t,t\in R^ 2_+$$, and sufficient conditions of their uniform integrability. It is mentioned that a nonnegative weak semimartingale is a local martingale iff $x_ t=\exp \{m_ t-2^{-1}<m+m^ 1>^ 1_ t-2^{-1}<m+m^ 2>^ 2_ t+2^{- 1}[m]_ t-[m,(m+m^ 1)*(m+m^ 2)]_ t\},$ where $$m^ i$$ are i- martingales satisfying some system of stochastic equations, $$<\cdot >^ 1_ t$$, $$<\cdot >^ 2_ t$$ and $$[\cdot]_ t$$ are “one-parameter” and “two-parameter” square functions. Let $E \exp \{\gamma^ 1_ t sh 2[m]_ t+\gamma^ 2_ t ch 2[m]_ t\}<\infty.$ where $$\gamma^ 1_ t=\sup_{s_ 1\leq t_ 1}<m>^ 2_ t$$, $$\gamma^ 2_ t=\sup_{s_ 2\leq t_ 2}<m>^ 1_ t$$. Then $$x_ t$$ is a martingale. Under the conditions $E \exp \{2^{-1}<m+m^ 1>^ 1_{\infty t_ 2}\}<\infty \quad and\quad E \exp \{2^{-1}<m+m^ 2>^ 2_{t_ 1\infty}\}<\infty$ $$x_ t$$ is uniformly integrable in $$t_ 1$$ under $$t_ 2$$ fixed and vice versa. The case when $$m_ t$$ is a strong martingale is also considered.
Reviewer: Yu.S.Mishura

### MSC:

 60G44 Martingales with continuous parameter 60G60 Random fields