Inequalities for vector-valued martingales with continuous time. (English. Ukrainian original) Zbl 0939.60025

Weisz, F.; Mishura, Yu. S.

Theory Probab. Math. Stat. 58, 9-25 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 8-23 (1998).

The authors consider vector-valued martingales with continuous time taking values in the space of unconditional martingale differences (UMD) with unconditional basis. Let \((\Omega,{\mathcal A},P)\) be a standard probability space and let \(({\mathcal F}_t,t\in R_+)\) be stochastic basis, that is: 1) if \(s<t\), then \({\mathcal F}_{s}\subset {\mathcal F}_{t}\subset {\mathcal A}\), 2) \({\mathcal F}_{t}=\bigcap_{s>t}{\mathcal F}_{s}\), 3) \(\bigcup_{t\in R_+}{\mathcal F}_{t}={\mathcal A}.\) Let \(X\) be an UMD space. The authors consider \(X\)-valued martingale \((M_{t},{\mathcal F}_{t})\) such that \(E\|M_{t}\|_{X}^{p}\leq C.\) It is proved that two versions of the maximal function and the quadratic variation exist. The authors obtain the Burkholder-Gundy inequalities for these two versions of maximal functions. It is constructed predicted martingale spaces for vector-valued martingale and atomic decompositions of martingales. These results are used for the investigation of the Davis decompositions for vector-valued martingales. The Davis inequality is obtained, too.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G44	Martingales with continuous parameter
60E15	Inequalities; stochastic orderings

Keywords:

martingale; Banach space; Burkholder-Gundy inequality; unconditional convergence; Davis inequality