Canonical representation of two-parameter strong semimartingales. (English. Russian original) Zbl 0629.60053

Mishura, Yu. S.

Theory Probab. Math. Stat. 33, 91-95 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 82-86 (1985).

Consider an increasing family of \(\sigma\)-fields \(\{F_ t\), \(t\in R^ 2_+\}\) verifying the usual conditions (1)-(4) of R. Cairoli and J. B. Walsh [Acta Math. 134, 111-183 (1975; Zbl 0334.60026)]. Strong semimartingales are introduced as two-parameter adapted processes \(\{\xi (t),t\in R^ 2_+\}\) satisfying the condition \[ \sup_{\pi}\sum_{\Delta \in \pi}E| E(\xi (\Delta)/\gamma_ s)| <\infty \] where \(\pi\) runs over all partitions of \(R^ 2_+\) into rectangles \(\Delta =[s,t]\), \(\xi\) (\(\Delta)\) denotes the increment of \(\xi\) on \(\Delta\), and \(\gamma_ s\) is the \(\sigma\)-field \(F_{s_ 1,\infty}\vee F_{\infty,s_ 2}\) if \(s=(s_ 1,s_ 2).\)
The main result of this paper is a decomposition of a strong semimartingale verifying some additional properties, into the sum of several components, one of them being a continuous local strong martingale, and the others being compensated sums of jumps or bounded variation random fields.

Reviewer: D.Nualart

MSC:

60G48	Generalizations of martingales
60G60	Random fields

Keywords:

Strong semimartingales; decomposition of a strong semimartingale; local strong martingale; random fields

Citations:

Zbl 0334.60026