Approximations of ruin probabilities in mixed Poisson models with lattice claim amounts

In this work we study uniform error bounds for approximations of the ruin probability in a mixed Poisson model with lattice claim amounts. The approximations considered are constructed by rounding the record lows up, down or to the midpoint. We observe how a round-to-the-midpoint approximation produces qualitatively better results than the other two methods, as in this case we can give error bounds which are uniformly bounded in the parameters of the model. We relate this approximation problem to an appropriate representation of the ruin probability in terms of Steklov operators. In our proofs, we use general results concerning the decrease of the Levy's concentration function for sums of independent and identically distributed random variables.