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

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

In Lewis (2001), he refers to the 'jump diffusion-case', when the underlying process is a Levy process with finite Levy measure - Which is consistent with the above.

Lewis, Alan L., A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes (September 2001). Available at SSRN: https://ssrn.com/abstract=282110 or http://dx.doi.org/10.2139/ssrn.282110

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

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