Multiparameter Poisson Processes and Martingales

We introduce and study a multiparameter Poisson process (MPP). In a particular case, it is observed that the MPP has a unique representation. Its subordination with the multivariate subordinator and inverse subordinator are studied in detail. Also, we consider a multivariate multiparameter Poisson process and establish its connection with the MPP. An integral of the MPP is defined, and its asymptotic distribution is obtained. Later, we study some properties of the multiparameter martingales. Moreover, the multiparameter martingale characterizations for the MPP and its subordinated variants are derived.