Cramér–Lundberg approximation for nonlinearly perturbed risk processes

An extension of the classical Cramér–Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér–Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations.