Volterra integral equations: An approach based on Lipschitz-continuity

In this study, we consider a linear Volterra integral equation of the second type whose unique unknown solution is known to be Lipschitz-continuous. Using this property, we derive a feasible, rapid, and accurate numerical algorithm. An application to risk theory is considered. More in detail in a Cramér-Lundberg model framework, using its integro-differential representation as a starting point, we prove the ruin probability to be a Lipschitz function. Using the proposed algorithm, we evaluate the ruin probability that solves the associated Volterra integral equation. To show that the proposed framework can be reasonably generalized, we considered a wide range of claim size distributions.