Some results on the joint distribution prior to and at the time of ruin in the classical model

For the classical risk model (i.e. with Poisson arrivals), we study the tail of the joint distribution of the surplus prior to and at ruin. In particular, we obtain some inequalities and monotonicity results for it. Let S be the random variable with distribution function the probability of non-ruin, 1−ψ(u), and the probability the surplus just before ruin exceeds x, given that ruin occurs. We estimate the distance between the residual lifetime of S, Pr(S>u+y∣S>u) and the product , where the tail convolution includes again the random variable S. Finally, based on this distance, we derive a lower bound for the probability of ruin, and we compare this against a bound available in the literature.