Martingale Approach to Derive Lundberg-Type Inequalities

In this paper, we find the upper bound for the tail probability Psupn⩾0∑I=1nξI>x with random summands ξ1,ξ2,… having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ1exp{−ϱ2x} with some positive constants ϱ1 and ϱ2. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.