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I am considering a portfolio of car insurance policies. In order to capture the individual history (driving skills, age, etc.) of policyholders, it is assumed that the claim numbers $N(t)$ are modeled by a mixed Poisson process, that is: $$N(t) := \hat{N}(\theta t), \quad t>0$$ where $\theta\sim\mathcal{E}(1)$ is the exponentially-distributed mixing variable, which is independent of the Poisson process $\hat{N}(t)$ and 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 $V_i\sim U(0,1)$ is uniformly-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 $t$, 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 want to compare it with the expected value of $S(t)$ (with respect to $N(t)$).

Thanks in advance.

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  • $\begingroup$ Do the claim sizes $X_i$ have a specific distribution? $\endgroup$ Oct 20 '20 at 11:50
  • $\begingroup$ No, but the $X_j$ are assumed to be i.i.d. and independent of both $T_j$ and $V_j$ $\endgroup$ Oct 20 '20 at 12:28
  • $\begingroup$ If they don't follow any specific distribution it will be hard to make a statement about $\bar{S}(t)$ that is more precise than you show, right? $\endgroup$
    – Bob Jansen
    Oct 27 '20 at 11:27

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