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

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.

**1.**N. V. Kartashov,*Strong stable Markov chains*, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR**1451375****2.**D. C. M. Dickson,*The probability of ultimate ruin with a variable premium rate*, Scand. Actuarial J. (1991), 75-86.**3.**Hans U. Gerber,*On the probability of ruin in the presence of a linear dividend barrier*, Scand. Actuar. J.**2**(1981), 105–115. MR**623405**, 10.1080/03461238.1981.10413735**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. 145-199.**5.**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**, 10.1214/aoap/1034625255**6.**G. C. Taylor,*Probability of ruin with variable premium rate*, Scand. Actuar. J.**2**(1980), 57–76. MR**578447**, 10.1080/03461238.1980.10408641**7.**M. V. Kartashov,*On ruin probabilities for a risk process with bounded reserves*, Teor. Ĭmovīr. Mat. Stat.**60**(1999), 46–58 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**60**(2000), 53–65 (2001). MR**1826141****8.**M. V. Kartashov and O. M. Stroēv,*The Lundberg approximation for the risk function in an almost homogeneous environment*, Teor. Ĭmovīr. Mat. Stat.**73**(2005), 63–71 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**73**(2006), 71–79. MR**2213842**, 10.1090/S0094-9000-07-00682-5**9.**M. V. Kartashov,*Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates*, Teor. Ĭmovīr. Mat. Stat.**78**(2008), 54–65 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**78**(2009), 61–73. MR**2446849**, 10.1090/S0094-9000-09-00762-5**10.**M. V. Kartashov,*Boundedness, limits, and stability of solutions of an inhomogeneous perturbation of a renewal equation on a half-line*, Teor. Ĭmovīr. Mat. Stat.**81**(2009), 65–75 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**81**(2010), 71–83. MR**2667311**, 10.1090/S0094-9000-2010-00811-8**11.**M. V. Kartashov,*A minimal uniform renewal theorem and transition phenomena for an inhomogeneous perturbation of a renewal equation*, Teor. Ĭmovīr. Mat. Stat.**82**(2010), 43–55 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**82**(2011), 27–41. MR**2790481**, 10.1090/S0094-9000-2011-00825-3**12.**N. V. Kartashov,*Uniform limit theorems for ergodic random processes and their applications in the queuing theory*, Doctoral Dissertation Thesis, Kiev, KGU, 1985. (Russian)**13.**N. V. Kartašov,*Power estimates for the rate of convergence in the renewal theorem*, Teor. Veroyatnost. i Primenen.**24**(1979), no. 3, 600–607 (Russian, with English summary). MR**541374****14.**N. V. Kartashov,*A generalization of Stone’s representation and necessary conditions for uniform convergence in the renewal theorem*, Teor. Veroyatnost. i Mat. Statist.**26**(1982), 49–62, 158 (Russian). MR**664903****15.**N. V. Kartashov,*Equivalence of uniform renewal theorems and their criteria*, Teor. Veroyatnost. i Mat. Statist.**27**(1982), 51–60, 158 (Russian). MR**673349****16.**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****17.**A. A. Borovkov,*Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya*, Izdat. “Nauka”, Moscow, 1972 (Russian). MR**0315800****18.**Charles Stone,*On absolutely continuous components and renewal theory*, Ann. Math. Statist.**37**(1966), 271–275. MR**0196795****19.**D. J. Daley,*Tight bounds for the renewal function of a random walk*, Ann. Probab.**8**(1980), no. 3, 615–621. MR**573298****20.**V. M. Shurenkov,*Ergodic theorems and related problems*, VSP, Utrecht, 1998. MR**1690361**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60J45,
60A05,
60K05

Retrieve articles in all journals with MSC (2010): 60J45, 60A05, 60K05

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/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