Advances in Applied Probability

Laplace transform identities for the volume of stopping sets based on Poisson point processes

Nicolas Privault

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We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

Article information

Adv. in Appl. Probab., Volume 47, Number 4 (2015), 919-933.

First available in Project Euclid: 11 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G57: Random measures 60G48: Generalizations of martingales 60H07: Stochastic calculus of variations and the Malliavin calculus

Poisson point process stopping set gamma-type distribution Girsanov identity anticipating stochastic calculus


Privault, Nicolas. Laplace transform identities for the volume of stopping sets based on Poisson point processes. Adv. in Appl. Probab. 47 (2015), no. 4, 919--933. doi:10.1239/aap/1449859794.

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