General bounds for the deficit distribution at ruin in the Sparre Andersen model

We present constructive methods that give upper and lower bounds of Ψ, the insurer’s deficit distribution at ruin time in the Sparre Andersen model. The methodology effectively provides monotone sequences of upper and lower bounds of Ψ, and those sequences converge to Ψ with exponential rate. Several new lower and upper bounds of Ψ are given. As an example, we introduce new bounds of the deficit distribution at ruin when the claim distribution satisfies the new better/worse than used property. We show that some of the examined lower and upper bounds of Ψ may coincide with the exact values of the considered probability.