= Independent compound Poisson jumps cannot cancel
Let $Y$ be a <Compound Poisson process> with nonzero jump marks and positive rate, <independent> of a <càdlàg> process $Z$. Conditional on the path of $Z$, its discontinuities form a <countable set>. The Poisson jump times have continuous distributions and avoid that set almost surely. At every jump of $Y$, the sum $Y+Z$ therefore has the same nonzero jump. In particular a <continuous Lévy process> cannot have a positive-rate <independent> compound Poisson component. Applying this to each truncation of its <Lévy measure> proves that its jump measure is zero.
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