If I have a compound possion process due to which I can generate a poison random measure $N$ then I can write my compound poisson process in the another form as $$\sum_{i=1}^{P_t}Y_i = \int_{\mathbb{R}\times [0,t]}zN(dz,ds).$$ Where $P_t$ is the poison process with intensity $\lambda$ and $Y_i$ are i.i.d jump sizes. If there is brownian motion $W$ independent of the jump sizes $Y_i$, and poisson process $P_t$ then what will be $$\mathbb{E}\Big[\exp\Big(\int_0^tf(s)dW(s)+ \int_{\mathbb{R}\times [0,t]}g(s,z)N(dz,ds)\Big) \Big] =\mathbb{E}\Big[\exp\Big(\int_0^tf(s)dW(s)\Big)\exp\Big( \int_{\mathbb{R}\times [0,t]}g(s,z)N(dz,ds)\Big) \Big] =? $$ where $f,g$ are deterministic continuous functions.
Can we just write it as a product of the expectation of two components or not? Some explanation or a proof would be greatly appreciated?