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