A rigorous definition of two-parameter point processes is given as a distribution of a denumerable number of random points in the plane. A characterization with stopping lines and relation with predictability are obtained. Using the one-parameter multivariate point-process representation, a general representation theorem for a wide class of martingales is presented, which extends the representation theorem with respect to a Poisson process.
Ely Merzbach. David Nualart. "A Martingale Approach to Point Processes in the Plane." Ann. Probab. 16 (1) 265 - 274, January, 1988. https://doi.org/10.1214/aop/1176991900