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

TL;DR

This paper derives closed-form formulas for the distribution of jumps in doubly-stochastic Poisson processes, using Bell polynomials and Malliavin calculus, with potential applications in credit risk modeling.

Contribution

It introduces two novel methods—Bell polynomial approach and Malliavin calculus—for obtaining explicit jump distribution formulas in doubly-stochastic Poisson processes.

Findings

Closed-form formulas expressed via Bell polynomials.

Recursive formulas derived using Malliavin calculus.

Potential applications in credit risk analysis.

Abstract

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 L\'evy 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.