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The stability of transient quasi-homogeneous Markov semigroups and an estimate of the ruin probability
Author(s):
M.
V.
Kartashov
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 75
(2006).
Journal:
Theor. Probability and Math. Statist.
No. 75
(2007),
41-50.
MSC (2000):
Primary 60J45;
Secondary 60A05
Posted:
January 23, 2008
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Additional information
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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- 1.
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht, The Netherlands/Kiev, Ukraine, 1996. MR 1451375 (99e:60150)
- 2.
- E. B. Dynkin, Theory of Markov Processes, Fizmatlit, Moscow, 1962; English transl., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0131900 (24:A1747)
- 3.
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- 4.
- T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966. MR 0203473 (34:3324)
- 5.
- 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)
- 6.
- 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)
- 7.
- 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)
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60J45,
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60A05
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
DOI:
10.1090/S0094-9000-08-00712-6
PII:
S 0094-9000(08)00712-6
Keywords:
Semigroup of operators,
infinitesimal operator,
uniform transiency,
strong stability
Received by editor(s):
19/DEC/2005
Posted:
January 23, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
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