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Pure jump process in Duffie, Pan and Singleton's paper

Essentially yes - $Z_t$ is a compound Poisson process, except that the underlying counting process $N_t$ has intensity $\lambda(X_t)$. I.e $$ N_t - N_s \sim Pois\bigg( \int_s^t \lambda(X_u) \mathrm{d}...
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