A note on Marked Point Processes and multivariate subordination

Abstract

The aim of this paper is to state a correspondence between marked Poisson processes and multivariate subordinated Lévy processes. We prove that, under suitable conditions, marked Poisson processes are in law subordinated Brownian motions and we provide their Lévy triplet and characteristic function. We introduce the class of multivariate Gaussian marked Poisson processes and prove that -in law -they are multivariate subordinated Brownian motions.

Multivariate Marked Poisson Processes and Market Related Multidimensional Information Flows

“…, d} in (2.1) is a multivariate process with correlated margins. The links between a multi-parameter process and Gaussian marks and between subordinator π(t) and the Poisson random measure Π are proved in Jevtić et al (2017). This connection provides a theoretical motivation to introduce multi-parameter Brownian motions as models of asset returns, by interpreting the multivariate subordinator π(t) as the whole information up to time t, measured by Π.…”

“…where Π 1i are the jump times and Π 2i are the jump sizes. The following theorem essentially proved in Jevtić et al (2017), provides a connection between marked Poisson processes and multivariate subordinated Lévy processes.…”

“…(2.6) By Theorem 1.1 in Jevtić et al (2017) the family N = (Π 1 , Z) forms a Poisson random measure on (R + × R n , B n+1 ) with mean measure µ N (dt, dy) = dt…”