Improvement of the stability of solutions of an inhomogeneous perturbed renewal equation on the semiaxis
Author:
M. V. Kartashov
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 84 (2012), 65-78
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/S0094-9000-2012-00854-5
Published electronically:
July 31, 2012
MathSciNet review:
2857417
Full-text PDF Free Access
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Additional Information
Abstract:
We consider a generalized inhomogeneous continuous-time 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.
References
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References
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, The Netherlands/Ukraine, 1996. MR 1451375 (99e:60150)
- D. C. M. Dickson, The probability of ultimate ruin with a variable premium rate, Scand. Actuarial J. (1991), 75–86.
- H. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuarial J. (1981), 105–115. MR 623405 (83c:62169)
- 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. 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)
- G. C. Tailor, Probability of ruin with variable premium rate, Scand. Actuarial J. (1980), 57–76. MR 578447 (81m:62188)
- N. V. Kartashov, On ruin probabilities for a risk process with bounded reserves, Theor. Probab. Math. Stat. 60 (2000), 46–58; English transl. in Theor. Probab. Math. Stat. 60 (2001), 53–65. MR 1826141
- 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), 64–72; 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. Veroyatnost. Mat. Statist. 78 (2008), 55–66; English transl. in Theor. Probab. Math. Stat. 78 (2009), 61–73. MR 2446849 (2010a:60295)
- 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), 65–75; English transl. in Theor. Probab. Math. Stat. 81 (2010), 71–83. MR 2667311 (2011f:60154)
- 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), 43–55; English transl. in Theor. Probab. Math. Stat. 82 (2011), 27–41. MR 2790481 (2011m:60264)
- N. V. Kartashov, Uniform limit theorems for ergodic random processes and their applications in the queuing theory, Doctoral Dissertation Thesis, Kiev, KGU, 1985. (Russian)
- N. V. Kartashov, Power estimates for the convergence rate in a renewal theorem, Teor. Veroyatnost. Primenen. 24 (1979) no. 3, 600–607; English transl. in Theory Probab. Appl. 24 (1980), no. 3, 606–612. MR 541374 (80i:60124)
- 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), 49–62; English transl. in Theor. Probab. Math. Stat. 26 (1983), 53–67. MR 664903 (83m:60113)
- N. V. Kartashov, Equivalence of uniform renewal theorems and their criteria, Teor. Veroyatnost. Mat. Statist. 27 (1982), 51–60; English transl. in Theor. Probab. Math. Stat. 27 (1984), 55–64. MR 673349 (83m:60114)
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, New York, 1966. MR 0210154 (35:1048)
- A. A. Borovkov, Stochastic Processes in Queueing Theory, Nauka, Moscow, 1972; English transl., Springer-Verlag, New York–Berlin, 1976. MR 0315800 (47:4349)
- C. Stone, On absolutely continuous components and renewal theory, Ann. Math. Stat. 37 (1966), 271–275. MR 0196795 (33:4981)
- D. J. Daley, Tight bounds for the renewal function of a random walk, Ann. Probab. 8 (1980), no. 3, 615–621. MR 573298 (81e:60094)
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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
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