On the deficit distribution when ruin occurs—discrete time model

The paper presents an algorithm of bounding from above and from below the deficit distribution at ruin. The algorithm is based on iterating a monotone integral operator which has a fixed point at the deficit distribution at ruin. The upper and lower bounds converge monotonically with an exponential rate to the exact value of the deficit distribution. A counterpart of the famous Cramér-Lundberg inequality for the distribution of the deficit at ruin is established. A general method to improve further the algorithm is given.