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

[摘要] We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by Lévy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.

[学科分类] 统计和概率

[关键词] Doubly-stochastic Poisson process;Bell polynomials;Malliavin calculus;credit risk;Hazard process;integrated non-Gaussian OU process