The moments of ruin time in the classical risk model with discrete claim size distribution

Firstly exact simple expressions are given for the moments Mr = E0(Tr 1{T<∞}) when the initial reserves are equal to zero. Then for positive initial reserves the same moments are expressed very compactly through the Mrʹs, and the polynomials en(t) = eλt P(St = n), n = 0, 1, … In both cases the results are exact and valid for any arithmetic claim size distribution. Our main tool is the generalized Appell structure of the polynomials family {en, n ∈ N}. Both cases c > λm and c ≤ λm are treated.