 Solution

Condition on $N$. Since the amounts are independent and identically distributed random variables and are independent of $N$, their moment-generating functions multiply:Summing against the positive-support geometric distribution gives the geometric-sum moment-generating functionThe finite-transform domain is $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. The terms are nonnegative for real $t$, so if the latter condition fails the sum diverges. For $t\le 0$ both conditions always hold, since the claims are positive; thus the same calculation always provides the Laplace transform of a nonnegative random variable. Positivity alone does not guarantee a finite moment-generating function on a positive neighbourhood of zero.

There is at least one strictly positive amount in the portfolio. Hence $\mathbb{P}(S=0)=0$; zero carries no atom of a measure to add to the continuous laws in the next two parts.