Electronic Communications in Probability

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

Arturo Valdivia

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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.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 13, 12 pp.

Received: 2 January 2017
Accepted: 22 February 2019
First available in Project Euclid: 21 March 2019

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Primary: 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; Lévy processes 60H07: Stochastic calculus of variations and the Malliavin calculus 91G40: Credit risk

Doubly-stochastic Poisson process Bell polynomials Malliavin calculus credit risk Hazard process integrated non-Gaussian OU process

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Valdivia, Arturo. Closed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processes. Electron. Commun. Probab. 24 (2019), paper no. 13, 12 pp. doi:10.1214/19-ECP221. https://projecteuclid.org/euclid.ecp/1553133702

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