The Davis inequalities and the Gundy decomposition for two-parameter strong martingales. I. (English. Russian original) Zbl 0749.60042
Theory Probab. Math. Stat. 42, 29-37 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 27-35 (1990).
Let \(M=(M_ t)_{t\in\mathbb{R}^ 2_ +}\) be a right continuous, two- parameter strong martingale such that \(\mathbb{E}\sup_{s\leq t}M_ s<\infty\) for all \(t=(t_ 1,t_ 2)\in\mathbb{R}^ 2_ +\). The authors prove that the quadratic variation \(([M]_ t)\) of \(M\) exists and that the Davis inequalities hold:
\[
C\mathbb{E}[M]_ t^{1/2}\leq\mathbb{E}\sup_{s\leq t} M_ s\leq D\mathbb{E}[M]_ t^{1/2}.
\]
Reviewer: E.Dettweiler (Reutlingen)
MSC:
60G44 | Martingales with continuous parameter |
60G48 | Generalizations of martingales |
60G17 | Sample path properties |