The Davis inequalities and the Gundy decomposition for two-parameter strong martingales. II. (English. Russian original) Zbl 0759.60057

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

Theory Probab. Math. Stat. 43, 65-76 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 59-69 (1990).

Summary: [For part I see ibid. 42, 29-37 (1991) resp. ibid. 42, 27-35 (1990; Zbl 0749.60042.]
Certain properties of stopping sets on the plane are studied. It is proved that the dual weakly predictable projection exists for nonintegrable random fields with bounded jumps, and it is shown that the weakly predictable projection of a jump of a strong martingale with integrable supremum is indistinguishable from zero. The inequality \(E(M_ t)^ 2\leq CE[M]_ t\) is obtained for a strong martingale \(M_ t\) with integrable supremum. (For part III see below).

MSC:

60G60	Random fields
60G48	Generalizations of martingales
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

stopping sets; weakly predictable projection; strong martingale; integrable supremum

Citations:

Zbl 0749.60042