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We study the obtainment of closed-form formulas for the distribution of thejumps of a doubly-stochastic Poisson process. The problem is approached in twoways. On the one hand, we translate the problem to the computation of multiplederivatives of the Hazard process cumulant generating function; this leads to aclosed-form formula written in terms of Bell polynomials. On the other hand,for Hazard processes driven by L\'evy processes, we use Malliavin calculus inorder to express the aforementioned distributions in an appealing recursivemanner. We outline the potential application of these results in credit risk.

Closed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processes