Do jumps in Levy processes need to be independent of the process itself?

I have a very basic question about Levy processes. Is the process of the form $$X_t=\sigma B_t + \sum_{i=1}^{N_t}\eta_i(X_t)$$ a Levy process? Here $$B_t$$ is a standard Brownian motion and $$N_t$$ is a standard Poisson process with intensity $$\lambda$$. For simplicity one can take $$\sigma(X_t)=\sigma$$ and $$\eta(X_t)$$ is a "nicely" behaved process (could be a deterministic function of $$X_t$$). In most of the examples I have seen the jump component is simply a compound Poisson Process, where the jumps are i.i.d., which is not the case here.

It seems to me that this would not be a Levy process. In particular, intuitively, the above process will not have independent increments as what happened up to some point $$t_1$$ will affect the distribution of the process between $$t_2$$ and $$t_1$$ via the dependence of the jumps (if they occur) on the position of the process. Is this intuition correct?

In general, are there any restrictions on the possible jump component of a stochastic processes that are needed for the process to be a Levy process?

• Yes, your intuition is correct. A process of the form $\sum_{i=1}^{N_t} \xi_i$ is a Lévy process iff $\xi_i$, $i \geq 1$, are iid random variables. – saz Apr 22 at 5:46
• Thank you for the clarification! – Mdoc Apr 22 at 17:06