The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution.

Assume the initial surplus is $U$, and the premium loading factor is $\theta$.

I am trying to find the probability of ruin for the first claim.

Here is what I have got:

Let $T$ denotes the time until the first claim and $T\sim Exp(\lambda)$, now I am looking for $\psi(U,T)$

and $\psi(U,T) = \Pr(U + (1+\theta)\lambda T < X_1)= \Pr\Big(T < \frac{X_1 - U}{(1+\theta)\lambda}\Big)$.

But now I am stuck, I am not sure how to continue with this expression.


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