For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion.
I want to compute the distribution of $z$. Can someone show me how to start?
I know that for a time interval $dt$, the probability for a jump is $\lambda dt$, but I don't know what to do with it.. I found here on page 26 that: $E[f(X_t)]= \lambda t \int f(y)G(x,dy)+(1-\lambda t)f(x)$ but I don't why this is valid?