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

Part of: Geometric probability and stochastic geometry Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  21 March 2016

Nicolas Privault
Nicolas Privault*
Affiliation:
Nanyang Technological University
*
∗ Postal address: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371. Email address: [email protected]
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Abstract

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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.

Keywords

Poisson point processstopping setgamma-type distributionGirsanov identityanticipating stochastic calculus

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G57: Random measures 60G48: Generalizations of martingales 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 919 - 933
Copyright
Copyright © Applied Probability Trust 2015 

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