The stability of transient quasi-homogeneous Markov semigroups and an estimate of the ruin probability
Author:
M. V. Kartashov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 75 (2007), 41-50
MSC (2000):
Primary 60J45; Secondary 60A05
DOI:
https://doi.org/10.1090/S0094-9000-08-00712-6
Published electronically:
January 23, 2008
MathSciNet review:
2321179
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Abstract:
A time nonhomogeneous semigroup of Markov operators in a Banach space is called quasi-homogeneous if its infinitesimal operator has a dense domain and can be represented as the sum of the infinitesimal operator of a homogeneous semigroup and a bounded operator function.
We obtain estimates of the strong stability of a nonhomogeneous semigroup for the case where the underlying homogeneous semigroup is uniformly transient.
References
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- M. V. Kartashov, On the stability of almost time-homogeneous Markov semigroups of operators, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 105–113 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 71 (2005), 119–128. MR 2144325, DOI https://doi.org/10.1090/S0094-9000-06-00652-1
- M. V. Kartashov, Ergodicity and stability of quasihomogeneous Markov semigroups of operators, Teor. Ĭmovīr. Mat. Stat. 72 (2005), 54–62 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 72 (2006), 59–68. MR 2168136, DOI https://doi.org/10.1090/S0094-9000-06-00664-8
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- Jan Grandell, Aspects of risk theory, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991. MR 1084370
References
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht, The Netherlands/Kiev, Ukraine, 1996. MR 1451375 (99e:60150)
- E. B. Dynkin, Theory of Markov Processes, Fizmatlit, Moscow, 1962; English transl., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0131900 (24:A1747)
- I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 2, “Nauka”, Moscow, 1973; English transl., Springer-Verlag, New York–Heidelberg, 1975. MR 0341540 (49:6288); MR 0375463 (51:11656)
- T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–New York, 1966. MR 0203473 (34:3324)
- M. V. Kartashov, On the stability of almost time-homogeneous Markov semigroups of operators, Teor. Imovir. Mat. Stat. 71 (2004), 105–113; English transl. in Theory Probab. Math. Statist. 72 (2005), 119–128. MR 2144325 (2006b:60172)
- M. V. Kartashov, Ergodicity and stability of quasihomogeneous Markov semigroups of operators, Teor. Imovir. Mat. Stat. 72 (2005), 54–62; English transl. in Theory Probab. Math. Statist. 72 (2006), 59–68. MR 2168136 (2006g:47065)
- M. V. Kartashov and O. M. Stroev, The Lundberg approximation for the risk function in an almost homogeneous environment, Teor. Imovir. Mat. Stat. 73 (2005), 63–71; English transl. in Theory Probab. Math. Statist. 73 (2006), 71–79. MR 2213842 (2007b:62121)
- J. Grandell, Aspects of Risk Theory, Springer Series in Statistics, Springer, 1991. MR 1084370 (92a:62151)
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Additional Information
M. V. Kartashov
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
winf@ln.ua
Keywords:
Semigroup of operators,
infinitesimal operator,
uniform transiency,
strong stability
Received by editor(s):
December 19, 2005
Published electronically:
January 23, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
