On the ruin probability of an insurance company dealing in a -market

Authors:
A. V. Baev and B. V. Bondarev

Translated by:
V. V. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **74** (2006).

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

Abstract | References | Similar Articles | 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.

**1.**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.**2.**P. A. Samuelson,*Proof that properly anticipated prices fluctuate randomly*, Industrial Management Review**6**(1965), 41-49.**3.**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****4.**A. V. Skorokhod,*Lectures on the theory of stochastic processes*, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR**1452108****5.**B. V. Bondarev,*Mathematical Models in Insurance*, Apeks, Donetsk, 2002. (Russian)**6.**I. I. Gikhman and A. V. Skorokhod,*\cyr Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya*, “Naukova Dumka”, Kiev, 1982 (Russian). MR**678374****7.**I. I. Gihman and A. V. Skorohod,*\cyr Stokhasticheskie differentsial′nye uravneniya*, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR**0263172**

Ĭ. Ī. Gīhman and A. V. Skorohod,*Stochastic differential equations*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR**0346904****8.**I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko,*Probability Theory and Mathematical Statistics*, Vyshcha Shkola, Kiev, 1988. (Russian)**9.**A. V. Baev and B. V. Bondarev,*An insurance company dealing in a market*, Applied statistics. Actuarial and Financial Mathematics (2003), no. 1-2, 11-26.**10.**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**

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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

DOI:
https://doi.org/10.1090/S0094-9000-07-00693-X

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