I would appreciate a hint regarding the following question (taken from Durret, Essentials of Stochastic Processes, questions 2.38
"Let $S_t$ be the price of stock at time t and suppose that at times of a Poisson process with rate the price is multiplied by a random variable $X_i$ > 0 with mean $\mu$ and variance $\sigma^2$. That is,
$S_t = S_0\prod_{i=1}^{N(t)}X_i$
where the product is 1 if $N(t) = 0$. Find ES(t)/ and varS(t).
Is this an example of thinning?
You start by the conditional expectation with respect to $N_t$.
$E[S_t|N_t] = S_0\prod_{i=1}^{N(t)} EX_i$ with the additional hypothesis that the process $N$ and the $X_i$s are independent, so
$$E[S_t|N_t] = S_0 \mu^{N(t)} $$
Let $r$ be the rate of the Poisson process, we have $$E[S_t] = EE[S_t|N_t] = S_0 \exp(-rt)\sum_{k=0}^\infty \mu^{k}\frac{(rt)^k}{k!} = S_0 \exp(rt(\mu-1))$$
We also have via the same computation: $$E[S_t^2] = E[S_0^2 (\sigma^2 + \mu^2)^N_t] = S_0^2 \exp(rt(\sigma^2 + \mu^2 - 1))$$
so $Var[S_t] = S_0^2 (\exp(rt(\sigma^2 + \mu^2 - 1)) - \exp(2rt(\mu-1)))$
One can actually compute the characteristic function of such a process (in terms of the characteristic function of the $X_i$s), this is a simple case of the Lévy–Khintchin representation.