P. Picard, et C. Lefèvre,, The probability of ruin in finite time with discrete claim size distribution, Scandinavian actuarial journal , 1997(1), 1997, pp. 58-69
The ruin time T is considered as the time of first crossing between a compound Poisson trajectory and an upper increasing boundary. Under the assumption that the claim sizes are integer-valued, we show that the distribution of T can be expressed in terms of generalized Appell polynomials. Using the algebraic properties of these polynomials elegant expressions are obtained for P(T > x).