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We consider a time inhomogeneous perturbation of the classical renewal equation with continuous time that can be reduced to the integral Volterra equation with a nonnegative bounded kernel. We assume that the kernel is approximated for large time intervals by a convolution kernel generated by a probability distribution. We prove that the limit of the solution of the perturbed equation exists if the corresponding perturbation of solutions of the perturbed equation is small. We consider an application for ruin functions of the classical risk process where the premium rate depends on the current capital of an insurance company. We obtain the exponential asymptotic behavior with the Lundberg index evaluated from the original (nonperturbed) intensity.

Inhomogeneous perturbations of a renewal equation and the Cramér–Lundberg theorem for a risk process with variable premium rates