Correlated Poisson processes and self-decomposable laws

We discuss in detail a procedure to produce two Poisson processes $M(t), N(t)$ associated to positively correlated, self-decomposable, exponential renewals. The main result of this paper is a closed, elementary form for the joint distribution $p_{m,n}(s,t)$ of the pair $(M(s), N(t))$: this turns out to be instrumental to produce explicit algorithms with applications to option pricing, as well as to credit and insurance risk modeling, that will be discussed in a separate paper