On the ruin probability of an insurance company dealing in a $BS$-market
Authors:
A. V. Baev and B. V. Bondarev
Translated by:
V. V. Semenov
Journal:
Theor. Probability and Math. Statist. 74 (2007), 11-23
MSC (2000):
Primary 60E15, 60H05, 60H30; Secondary 62P05
DOI:
https://doi.org/10.1090/S0094-9000-07-00693-X
Published electronically:
June 25, 2007
MathSciNet review:
2336774
Full-text PDF Free Access
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Additional Information
Abstract: We study a mathematical model of an insurance company that shares its capital by investing it in both stocks and bonds. The basic tool to describe the evolution of the stock price is the Ornstein–Uhlenbeck process. We construct an estimate for the ruin probability of an insurance company as a function of the initial capital. The distribution of the capital between stocks and bonds is found for which this estimate is minimal.
References
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- P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6 (1965), 41–49.
- Albert N. Shiryaev, Essentials of stochastic finance, Advanced Series on Statistical Science & Applied Probability, vol. 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. Facts, models, theory; Translated from the Russian manuscript by N. Kruzhilin. MR 1695318
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- B. V. Bondarev, Mathematical Models in Insurance, Apeks, Donetsk, 2002. (Russian)
- I. I. Gikhman and A. V. Skorokhod, Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374
- I. I. Gihman and A. V. Skorohod, Stokhasticheskie differentsial′nye uravneniya, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR 0263172
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics, Vyshcha Shkola, Kiev, 1988. (Russian)
- A. V. Baev and B. V. Bondarev, An insurance company dealing in a $(B,S)$ market, Applied statistics. Actuarial and Financial Mathematics (2003), no. 1–2, 11–26.
- Alexander Melnikov, Risk analysis in finance and insurance, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 131, Chapman & Hall/CRC, Boca Raton, FL, 2004. Translated from the Russian manuscript and edited by Alexei Filinkov. MR 2013235
References
- A. V. Baev and B. V. Bondarev, Ornstein–Uhlenbeck process and its applications in problems of actuarial mathematics, Applied statistics. Actuarial and Financial Mathematics (2002), no. 1, 3–28.
- P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6 (1965), 41–49.
- A. N. Shiryaev, Essentials of Stochastic Finance. Facts, Models, Theory, Fazis, Moscow, 1998; English transl., World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1695318 (2000e:91085)
- A. V. Skorokhod, Lectures on the Theory of Stochastic Processes, Lybid’, Kyiv, 1990; English transl., VSP/TViMS, Utrecht, Netherlands/Kiev, Ukraine, 1996. MR 1452108 (99d:60001)
- B. V. Bondarev, Mathematical Models in Insurance, Apeks, Donetsk, 2002. (Russian)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications, Naukova dumka, Kiev, 1982. (Russian) MR 678374 (84j:60003)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, 1968; English transl., Springer-Verlag, New York–Heidelberg, 1972. MR 0263172 (41:7777); MR 0346904 (49:11625)
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics, Vyshcha Shkola, Kiev, 1988. (Russian)
- A. V. Baev and B. V. Bondarev, An insurance company dealing in a $(B,S)$ market, Applied statistics. Actuarial and Financial Mathematics (2003), no. 1–2, 11–26.
- A. V. Melnikov, Risk Analysis in Finance and Insurance, Ankil, Moscow, 2001; English transl., Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2013235 (2004i:91004)
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Additional Information
A. V. Baev
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk State University, Universitets’ka Street 24, 83055 Donetsk, Ukraine
Email:
tv@matfak.dongu.donetsk.ua
B. V. Bondarev
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk State University, Universitets’ka Street 24, 83055 Donetsk, Ukraine
Email:
bvbondarev@cable.netlux.org
Keywords:
Poisson measure,
stochastic integral,
investor’s portfolio,
ruin probability
Received by editor(s):
January 11, 2005
Published electronically:
June 25, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
