General bounds for infinite-horizon insolvency risk measures in the Sparre Andersen model

We apply an integral risk operator L (which generalizes Taylor’s Volterra-type operator K for ruin probabilities in a classical Cramér–Lundberg model) to obtain new lower and upper bounds of more general infinite-horizon insolvency risk measures δ in the Sparre Andersen risk model. We prove that these measures satisfy the fixed-point integral equation and that L is monotone. Based upon these properties, we introduce constructive and effective methodologies that give the bounds of δ (considered as a properly chosen solution of an appropriate fixed-point equation) by means of suitably adjusted operator bounds. These bounds may coincide with the exact values of the explored risk measures. In addition, we present how to handle with a problem of infinitely many fixed points of L. The introduced methods are applicable for a variety of significant insolvency risk measures used in insurance mathematics and mathematical finance in discrete- and continuous-time frameworks. As an example, we introduce new upper and lower operator bounds of δ , including a generalization and improvement of the Cramér–Lundberg bound. We also give an algorithm to approximate δ by a numerical fixed point of L.