Coupling Poisson processes by self-decomposability

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 (M(s), N(t)), an outcome which turns out to be instrumental to produce explicit algorithms for future applications. We finally discuss the cross-correlation properties of the two processes and the relative timing of their jump