In page 1349 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 $\{X_s:0\le{s}<T\}$ 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 $Pois(\lambda(X_t))$.

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 $\{X_s:0\le{s}<T\}$ 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 $Pois(\lambda(X_t))$.

In page 1349 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 $\{X_s:0\le{s}<T\}$ 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 $Pois(\lambda(X_s))$.

In page 1349 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 $\{X_s:0\le{s}<T\}$ 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 $Pois(\lambda(X_t))$.