# Poisson modelling of non-life insurance claims with reporting delay

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 $$N(t)$$).

• Do the claim sizes $X_i$ have a specific distribution? Oct 20 '20 at 11:50
• No, but the $X_j$ are assumed to be i.i.d. and independent of both $T_j$ and $V_j$ Oct 20 '20 at 12:28
• 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? Oct 27 '20 at 11:27