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

In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" the pure jump process Z is defined as follow:

conditional on the path of X, the jump times of Z are the jump times of a Poisson process with time-varying intensity $$\{\lambda(X_s):0\le{s}\le{t}\}$$, and that the size of the jump of Z at a jump time T is independent of $$\{X_s:0\le{s} and has the probability distribution $$\nu$$.

Since I can't find a formal definition of Z in the same paper, I'm writing to ask if it is talking about a compound Poisson Process. Saying, $$(Z_t)_{t\ge{0}}$$ is a stochastic process such that: $$Z_t=\sum_{i=1}^{N_t}{Y_i}$$, where $$Y_1,...,Y_{N_t}$$ are random variables with distribution $$\nu$$ and $$N_t$$ follows a $$Pois(\lambda(X_t))$$.

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