Davis inequalities and the Gundy decomposition for two-parametric strong martingales. III. (Russian) Zbl 0759.60058

Gushchin, A. A.; Mishura, Yu. S.

Teor. Veroyatn. Mat. Stat., Kiev 44, 49-56 (1991).

The paper contains two principal results from the theory of two-parameter martingales. The first result is that the strong martingale \(M\) with integrable supremum \((M\in M^*)\) and bounded jumps \(\square M_ t\) is locally square-integrable \((M\in M^ 2_{s,\text{loc}})\). The second one is that under some additional assumptions the strong martingale \(M\in M^*\) admits the decomposition \(M=M^ 1+M^ 2\), where \(M^ 1\in M^ 2_{s,\text{loc}}\cap M^*_ s\), \(M^ 2\in M^*_ s\) and has bounded variation (two-parameter version of Gundy’s decomposition for martingales). The proofs are based on the results of parts I [Theory Probab. Math. Stat. 42, 29-37 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 27-35 (1990; Zbl 0749.60042)] and II [see the paper, reviewed above].

Reviewer: Yu.S.Mishura (Kiev)

MSC:

60G60	Random fields
60G44	Martingales with continuous parameter

Keywords:

two-parameter version of Gundy’s decomposition for martingales; two- parameter martingales; integrable supremum

Citations:

Zbl 0749.60042