Boundedness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis
Author:
M. V. Kartashov
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 71-83
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/S0094-9000-2010-00811-8
Published electronically:
January 18, 2011
MathSciNet review:
2667311
Full-text PDF Free Access
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Additional Information
Abstract:
We consider a time-nonhomogeneous perturbation of the classical renewal equation with continuous time on the semiaxis that can be reduced to the Volterra integral equation with a nonnegative bounded (or with a substochastic) kernel. We assume that this kernel is approximated by a convolution kernel for large time intervals and that the latter is generated by a substochastic distribution. We find necessary and sufficient conditions for the existence of the limit of a solution of the perturbed equation under the assumption that the corresponding perturbation is small in a certain sense. We also obtain estimates for the deviation of the solutions of the perturbed equations from those of the initial equations.
Several applications are described.
References
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References
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- D. C. M. Dickson, The probability of ultimate ruin with a variable premium rate, Scand. Actuar. J. (1991), 75–86.
- H. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuar. J. (1981), 105–115. MR 623405 (83c:62169)
- H. Gerber and E. S. W. Shiu, On the time value of ruin, Proc. of the 31 Actuarial Research Conference, Ball Statte Univ., Aug. 1996, pp. 145–199.
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- N. V. Kartashov, Uniform limit theorems for ergodic random processes and their applications in the queueing theory, Doctoral Dissertation, Kiev University, Kiev, 1985. (Russian)
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Additional Information
M. V. Kartashov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
nkartashov@skif.com.ua
Keywords:
Volterra equation,
renewal theorem,
transition kernel,
regularity,
minimal solution,
stability
Received by editor(s):
October 12, 2009
Published electronically:
January 18, 2011
Article copyright:
© Copyright 2010
American Mathematical Society
