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Theory of Probability and Mathematical Statistics

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

Author: M. V. Kartashov
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 61-73
MSC (2000): Primary 60J45; Secondary 60A05, 60K05
Published electronically: August 4, 2009
MathSciNet review: 2446849
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Abstract: 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.

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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Keywords: Volterra equation, renewal theorem, risk function, Lundberg index, Poisson process
Received by editor(s): January 15, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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