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

Mishura, Yu. S.

Teor. Veroyatn. Mat. Stat., Kiev 37, 94-105 (1987).

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

Keywords:

non-negative martingale; exponential representations for two-parameter continuous martingales; uniform integrability; strong martingale