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 pm,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