Multiparameter Poisson Processes and Martingales

TL;DR

This paper introduces and analyzes a multiparameter Poisson process (MPP), exploring its representations, subordinations, connections with other processes, and properties of associated martingales, including asymptotic behaviors.

Contribution

It provides a comprehensive study of MPP, including its unique representation, subordination, connections with multivariate processes, and martingale characterizations, advancing the theoretical understanding of multiparameter stochastic processes.

Findings

Unique representation of MPP established

Asymptotic distribution of the integral of MPP derived

Martingale characterizations for MPP and its variants obtained

Abstract

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.