Multiparameter Poisson Processes and Martingales
P. Vishwakarma, K. K. Kataria
TL;DR
The paper analyzes a multiparameter Poisson process (MPP) indexed by R^d_+ and shows it can be represented as the sum of independent one-parameter Poisson processes, establishing its additive Lévy process structure. It then studies time-changed variants via multivariate subordinators and inverse subordinators, deriving explicit distributions, Laplace transforms, and generator relations for steady-state and fractional dynamics, including a multivariate stable subordinator and inverse stable times. The authors introduce and characterize the MFPP, a finite sum of independent fractional Poisson processes, along with its autocovariance and long-range dependence properties via Mittag-Leffler functions, and provide fractional differential equations governing their evolution. Finally, the paper furnishes multiparameter martingale characterizations for the MPP and MFPP, connecting martingale properties to the underlying Poisson and fractional Poisson structures and enriching the stochastic-analytic toolbox for multiparameter processes.
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.