Improvement of the stability of solutions of an inhomogeneous perturbed renewal equation on the semiaxis
Author:
M. V. Kartashov
Translated by:
O. I. Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 84 (2011).
Journal:
Theor. Probability and Math. Statist. 84 (2012), 6578
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
Published electronically:
July 31, 2012
MathSciNet review:
2857417
Fulltext PDF
Abstract 
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Additional Information
Abstract: We consider a generalized inhomogeneous continuoustime renewal equation on the semiaxis that reduces to the Volterra integral equation with a nonnegative bounded (or substochastic) kernel. It is assumed that the kernel can be approximated in a large time scale by a convolution kernel generated by a stochastic distribution. Under some asymptotic assumptions imposed on the perturbation we find an improved condition for the boundedness; the latter condition is used to prove the existence of the limit of a solution of the perturbed equation and to establish estimates for the deviation from a solution of a nonperturbed equation. Some examples are discussed.
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 N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, The Netherlands/Ukraine, 1996. MR 1451375 (99e:60150)
 2.
 D. C. M. Dickson, The probability of ultimate ruin with a variable premium rate, Scand. Actuarial J. (1991), 7586.
 3.
 H. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuarial J. (1981), 105115. MR 623405 (83c:62169)
 4.
 H. Gerber and E. S. W. Shiu, On the time value of ruin, Proc. of the 31st Actuarial Research Conference, Ball State Univ., Aug. 1996, pp. 145199.
 5.
 H. Schmidli, An extension to the renewal theorem and an application to risk theory, Ann. Appl. Probab. 7 (1997), no. 1, 121133. MR 1428752 (97k:60234)
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 G. C. Tailor, Probability of ruin with variable premium rate, Scand. Actuarial J. (1980), 5776. MR 578447 (81m:62188)
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 N. V. Kartashov, On ruin probabilities for a risk process with bounded reserves, Theor. Probab. Math. Stat. 60 (2000), 4658; English transl. in Theor. Probab. Math. Stat. 60 (2001), 5365. MR 1826141
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 M. V. Kartashov and O. M. Stroyev, The Lundberg approximation for the risk function in an almost homogeneous environment, Teor. Imovir. Mat. Stat. 73 (2005), 6472; English transl. in Theor. Probab. Math. Stat. 73 (2006), 7179. MR 2213842 (2007b:62121)
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 M. V. Kartashov, Inhomogeneous perturbations of a renewal equation and the CramérLundberg theorem for a risk process with variable premium rates, Teor. Veroyatnost. Mat. Statist. 78 (2008), 5566; English transl. in Theor. Probab. Math. Stat. 78 (2009), 6173. MR 2446849 (2010a:60295)
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 M. V. Kartashov, Boundedness, limits, and stability of solutions of renewal equation with a nonhomogeneous perturbation on the semiaxis, Teor. Veroyatnost. Mat. Statist. 81 (2009), 6575; English transl. in Theor. Probab. Math. Stat. 81 (2010), 7183. MR 2667311 (2011f:60154)
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 M. V. Kartashov, A minimal uniform renewal theorem and transition phenomena for a nonhomogeneous perturbation of the renewal equation, Teor. Veroyatnost. Mat. Statist. 82 (2010), 4355; English transl. in Theor. Probab. Math. Stat. 82 (2011), 2741. MR 2790481 (2011m:60264)
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 N. V. Kartashov, Uniform limit theorems for ergodic random processes and their applications in the queuing theory, Doctoral Dissertation Thesis, Kiev, KGU, 1985. (Russian)
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 N. V. Kartashov, Power estimates for the convergence rate in a renewal theorem, Teor. Veroyatnost. Primenen. 24 (1979) no. 3, 600607; English transl. in Theory Probab. Appl. 24 (1980), no. 3, 606612. MR 541374 (80i:60124)
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 N. V. Kartashov, A Generalization of the Stone representation and necessary conditions for uniform convergence in the renewal theorem, Teor. Veroyatnost. Mat. Statist. 26 (1982), 4962; English transl. in Theor. Probab. Math. Stat. 26 (1983), 5367. MR 664903 (83m:60113)
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 N. V. Kartashov, Equivalence of uniform renewal theorems and their criteria, Teor. Veroyatnost. Mat. Statist. 27 (1982), 5160; English transl. in Theor. Probab. Math. Stat. 27 (1984), 5564. MR 673349 (83m:60114)
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 W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, New York, 1966. MR 0210154 (35:1048)
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 D. J. Daley, Tight bounds for the renewal function of a random walk, Ann. Probab. 8 (1980), no. 3, 615621. MR 573298 (81e:60094)
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 V. M. Shurenkov, Ergodic Theorems and Related Problems, Nauka, Moscow, 1989; VSP, Utrecht, 1998. MR 1690361 (2000i:60002)
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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 4E, Kiev 03127, Ukraine
Email:
nkartashov@skif.com.ua
DOI:
http://dx.doi.org/10.1090/S009490002012008545
Keywords:
Volterra equation,
renewal theory,
transition kernel,
minimal solution,
stability
Received by editor(s):
September 3, 2010
Published electronically:
July 31, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
