The excursion measure, product intensity, Poisson box counts and stopped-past step compensation.

The imported modules prove the Laplace functional, general and left-continuous predictable compensation, and full support identification. Support transfers through measurable compact-window lower bounds; the general predictable identity follows from finite-measure uniqueness.

References

K. Itô, Poisson point processes attached to Markov processes, Proc. Sixth Berkeley Symp. 3 (1972), 225–239; J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter IV; O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002, Chapters 12 and 22.