Some inequalities for the risk function in the time and space nonhomogeneous Cramér-Lundberg risk model
Authors:
M. V. Kartashov and V. V. Golomozyĭ
Translated by:
N. N. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal:
Theor. Probability and Math. Statist. 98 (2019), 243-254
MSC (2010):
Primary 91B30; Secondary 60J25
DOI:
https://doi.org/10.1090/tpms/1074
Published electronically:
August 19, 2019
MathSciNet review:
3824690
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Abstract | References | Similar Articles | Additional Information
Abstract: We consider a generalized Cramér-Lundberg risk model with a time nonhomogeneous Poisson process of insurance claims where the intensity of premiums depends on the current insurance capital. Some explicit bounds are found for the exponential normalized uniform distance between
- (a)
- the risk function in a time nonhomogeneous model and that in a time homogeneous but space nonhomogeneous model,
- (b)
- the risk functions in a time homogeneous model or in a space nonhomogeneous model and that in a time and space homogeneous model.
- [1] Evgenii B. Dynkin and Aleksandr A. Yushkevich, Markov processes: Theorems and problems, Translated from the Russian by James S. Wood, Plenum Press, New York, 1969. MR 0242252
- [2] Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. II, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by Samuel Kotz; Die Grundlehren der Mathematischen Wissenschaften, Band 218. MR 0375463
- [3] Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024
- [4] V. M. Shurenkov, Èrgodicheskie protsessy Markova, Teoriya Veroyatnosteĭ i Matematicheskaya Statistika [Probability Theory and Mathematical Statistics], vol. 41, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1087782
- [5] Jan Grandell, Aspects of risk theory, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991. MR 1084370
- [6] Edward W. Frees, James C. Hickman: an actuary who made a difference, N. Am. Actuar. J. 11 (2007), no. 1, 1–16. With notes in memory of Hickman. MR 2345807, https://doi.org/10.1080/10920277.2007.10597429
- [7] S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609
- [8] N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- [9] 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, https://doi.org/10.1090/S0094-9000-06-00652-1
- [10] 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, https://doi.org/10.1090/S0094-9000-06-00664-8
- [11] 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, https://doi.org/10.1090/S0094-9000-07-00682-5
- [12] M. V. Kartashov, Stability of transient quasihomogeneous Markov semigroups and an estimate for ruin probability, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 36–44 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 41–50. MR 2321179, https://doi.org/10.1090/S0094-9000-08-00712-6
- [13] 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, https://doi.org/10.1090/S0094-9000-2010-00811-8
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Additional Information
M. V. Kartashov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
nkartashov@skif.com.ua
V. V. Golomozyĭ
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
vitaliy.golomoziy@gmail.com
DOI:
https://doi.org/10.1090/tpms/1074
Keywords:
Time and space nonhomogeneous Markov processes,
ruin probability,
Cram\'er--Lundberg risk model
Received by editor(s):
November 2, 2017
Published electronically:
August 19, 2019
Additional Notes:
M. V. Kartashov passed away on January 1, 2018. See the tribute to him by the Theory of Probability and Mathematical Statistics Editorial Board in this issue
Article copyright:
© Copyright 2019
American Mathematical Society