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.
- H. Schmidli, An extension to the renewal theorem and an application to risk theory, Ann. Appl. Probab. 7 (1997), no. 1, 121–133. MR 1428752 (97k:60234)
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- M. V. Kartashov and O. M. Stroev, The Lundberg approximation for the ruin function in an almost homogeneous environment, Teor. Ĭmovir. Mat. Stat. 73 (2005), 63–71; English transl. in Theor. Probab. Math. Stat. 73 (2006), 71–79. MR 2213842 (2007b:62121)
- M. V. Kartashov, Inhomogeneous perturbations of a renewal equation and the Cramér–Lundberg theorem for a risk process with variable premium rates, Teor. Ĭmovir. Mat. Stat. 78 (2008), 55–66; English transl. in Theor. Probab. Math. Stat. 78 (2009), 61–73. MR 2446849 (2010a:60295)
- 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