Correlated Poisson processes and self-decomposable laws

TL;DR

This paper introduces a method to generate correlated Poisson processes using self-decomposable exponential renewals, providing explicit joint distributions and copulas, with applications in finance and queuing theory.

Contribution

It presents a novel family of copulas for correlated Poisson processes and derives their joint distributions in closed form, enabling practical simulation and analysis.

Findings

Derived explicit joint distribution formulas for correlated Poisson processes.

Identified copulas that couple the renewal processes with positive correlation.

Discussed the cross-correlation and timing properties of the processes.

Abstract

We analyze a method to produce pairs of non independent Poisson processes $M(t),N(t)$ from positively correlated, self-decomposable, exponential renewals. In particular the present paper provides the family of copulas pairing the renewals, along with the closed form for the joint distribution $p_{m,n}(s,t)$ of the pair $\big(M(s),N(t)\big)$, an outcome which turns out to be instrumental to produce explicit algorithms for applications in finance and queuing theory. We finally discuss the cross-correlation properties of the two processes and the relative timing of their jumps