A martingale approach to point processes in the plane
This paper extends some of the martingale theory of one-dimensional point processes to the two-dimensional case. The case considered is that in which no horizontal or vertical line has more than one point of the process, the F4 conditions hold for the filtration and the filtration is that generated by the process. The approach used, as in the reviewer, \textit{B. G. Ivanoff} and \textit{N. C. Weber} [Stochastic processes, Appl. 23, 307-318 (1986; Zbl 0614.60046)] is to consider the two-dimensional point process as a sequence of random points along the x-axis with random marks along the y-axis or vice-versa. An integral representation theorem is then proved, analogous to the 1- dimensional theorem of \textit{J. Jacod} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 31, 235-253 (1975; Zbl 0302.60032)]. The theorem assumes additionally that the 1- and 2-dual predictable projections are continuous. It would appear from Brown, Ivanoff and Weber, Lemma 3.1, that the assumption of continuity implies that a-priori assumption of single points on horizontal or vertical lines can be discarded.
martingale theory of one-dimensional point processes
scientific article; zbMATH DE number 4042970