On some properties of two-parameter martingales with jumps. (Russian) Zbl 0577.60044

Mishura, Yu. S.

Teor. Veroyatn. Mat. Stat. 29, 72-84 (1983).

Let \(\xi\) be a quadratically integrable two-parameter martingale, the trajectories of which belong to D with probability 1. Let \(\sup_{t}E\xi^ 2(t))<\infty\) and \(\xi^ 2(t)\) be regular with respect to both coordinates. Under a certain additional condition it is shown, that the random field \(\nu_{\xi}(t,A)\), namely the number of jumps of \(\xi\) on [0,t] with values in A, is a strong regular submartingale and admits a certain unique decomposition.
Furthermore, \(\xi\) may be represented as the sum of four orthogonal, quadratically integrable martingales. One of these is a pure ”jump” process, two are ”semi-continuous”, and one is continuous with probability 1. Sufficient conditions for the existence of the quadratic variation of \(\xi\) are given.

Reviewer: C.Baldauf

MSC:

60G44	Martingales with continuous parameter
60G60	Random fields

Keywords:

two-parameter martingale; strong regular submartingale; existence of the quadratic variation