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), 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

Abstract | References | Similar Articles | 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.

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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:
https://doi.org/10.1090/S0094-9000-2012-00854-5

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