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Random time change and an integral representation for marked stopping times
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  • Published: June 1990

Random time change and an integral representation for marked stopping times

  • Olav Kallenberg1 

Probability Theory and Related Fields volume 86, pages 167–202 (1990)Cite this article

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Summary

Consider the set\(C\) of all possible distributions of triples (τ, κ, η), such that τ is a finite stopping time with associated mark κ in some fixed Polish space, while η is the compensator random measure of (τ, κ). We prove that\(C\) is convex, and that the extreme points of\(C\) are the distributions obtained when the underlying filtration is the one induced by (τ, κ). Moreover, every element of\(C\) has a corresponding unique integral representation. The proof is based on the peculiar fact that EV τ, κ=0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT τ, κ isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of ζ, a suitably discounted version of η, is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.

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Author information

Authors and Affiliations

  1. Division of Mathematics, 120 Math. Annex, Auburn University, 36849, AL, USA

    Olav Kallenberg

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  1. Olav Kallenberg
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Research supported by NSF grant DMS-8703804

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Cite this article

Kallenberg, O. Random time change and an integral representation for marked stopping times. Probab. Th. Rel. Fields 86, 167–202 (1990). https://doi.org/10.1007/BF01474641

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  • Received: 03 July 1989

  • Revised: 29 December 1989

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01474641

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Keywords

  • Lebesgue Measure
  • Integral Representation
  • Extreme Point
  • Mathematical Biology
  • Point Process
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