Exact non-ruin probabilities in arithmetic case

Using the Wiener-Hopf method, for the model with arithmetic distributions of waiting times $T_i$ and claims $Z_i$ in ordinary renewal process, an exact non-ruin probabilities for an insurance company in terms of the factorization of the symbol of the discrete Feller-Lundberg equation, are obtained. The delayed stationary process is introduced and generating function for delay is given. It is proved that the stationary renewal process in arithmetic case is ordinary if and only if, when the inter-arrival times have the shifted geometrical distribution. A formula for exact non-ruin probabilities in delayed stationary process is obtained. Illustrative examples when the distributions of $T_i$ and $Z_i$ are shifted geometrical or negative binomial with positive integer power are considered. In these cases the symbol of the equation is rational functions what allows us to obtain the factorization in explicit form.