I am considering a portfolio of car insurance policies. In order to capture the individual history (driving skills, age,...) of policyholders it is assumed that the claim numbers $N(t)$ are modelled by a mixed Poisson process, that is $N(t) = \hat{N}(\theta t)$ $t>0$, where $\theta\sim$ Exp(1) is the mixing variable, which is independent of a Poisson process $\hat{N}(t)$; $t>0$ with intensity $1$.
However, in my case, general claims in an insurance portfolio are not reported at the arrival times $T_i$ but at times $T_i + V_i$ with a delay $V_i > 0$. I assume that $V_i\sim U(0,1)$ distributed.
An example for such delays is a policyholder who is injured in a car accident and does not have the opportunity to call his agent, immediately.
So I derive the number of claims reported up to time $t$ is given by,
$$\bar{N}(t)=\sum_{j=1}^{N(t)}1_{[0,t]}(T_j+V_j)$$
I assume that $V_j$ is independent of $T_j$ for each $j$.
I assume that the claim sizes $X_i$ are I.I.D and I want to determine the expected total claim amount $\bar{S}(t)$, that is $\mathbb{E}[\bar{S}(t)]$, where,
$$\bar{S}(t)=\sum_{j=1}^{\bar{N}(t)}X_j$$.
Furthermore, I also like to compare it with the expected value of $S(t)$ (with respect to $N(t)$).
Thanks in advance.