Summary
Let\(\hat L\) be a continuous additive functional with supportC of a Hunt processX={X t;t≧0}. LetS={S t;t≧0} be the inverse of\(\hat L\) and put\(Y_t = X_{S_t } \). For each time of discontinuityu ofS, letZ u be the corresponding excursion ofX outside ofC. The conditional structure of the excursion process {Z u;u≧0} given the paths ofY={Y t;t≧0} is studied. It is shown that conditionally, givenY, the excursion process is a Poisson random measure.
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Support from the Office of Naval Research (Contract Number N-00014-67-A-0112-0011) and the National Science Foundation (Grant Number ENG 75-02026) is gratefully acknowledged
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Jacobs, P.A. Excursions of a Markov process induced by continuous additive functionals. Z. Wahrscheinlichkeitstheorie verw Gebiete 44, 325–336 (1978). https://doi.org/10.1007/BF01013195
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DOI: https://doi.org/10.1007/BF01013195