An estimate for the ruin probability in a model with variable premiums and with investments in a bond and several stocks

Author:
M. O. Androshchuk

Translated by:
O. I. Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **76** (2007).

Journal:
Theor. Probability and Math. Statist. **76** (2008), 1-13

MSC (2000):
Primary 60H05; Secondary 60G15

DOI:
https://doi.org/10.1090/S0094-9000-08-00726-6

Published electronically:
July 10, 2008

MathSciNet review:
2368734

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a risk process generalizing the classical Cramér-Lundberg process. The feature of the process is that its price function depends on the current reserve of an insurance company as well as on its portfolio consisting of a riskless bond and a finite number of risky assets, modeled by geometric Brownian motions. We obtain an analogue of the classical exponential estimate for the ruin probability in this case. It turns out that the estimate for the model with investments is better than the corresponding estimate for the classical model without investments.

**1.**J. Gaier, P. Grandits, and W. Schachermayer,*Asymptotic ruin probabilities and optimal investment*, Ann. Appl. Probab.**13**(2003), 1054-1076. MR**1994044 (2004k:91124)****2.**Q. Tang,*Asymptotic Ruin Probabilities of the Renewal Model with Constant Interest Force and Regular Variation*, Technical Report, No. 9/04, 2004. MR**2118521 (2005k:62241)****3.**J. Cai and D. C. M. Dickson,*Upper Bounds for Ultimate Ruin Probabilities in the Sparre Andersen Model with Interest*, Research paper number 97, Melbourne, Insurance Math. Econom.**32**(2003), 61-71. 2002. MR**1958769 (2003k:60046)****4.**J. Grandell,*Aspects of Risk Theory*, Springer-Verlag, New York, 1991. MR**1084370 (92a:62151)****5.**C. Kluppelberg, A. E. Kyprianou, and R. A. Maller,*Ruin probabilities and overshoots for central Lévy insurance risk processes*, Ann. Appl. Probab.**14**(2004), 1766-1801. MR**2099651 (2005j:60097)****6.**J. Paulsen, J. Kasozi, and A. Steigen,*A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments*, Insurance Math. Econom.**36**(2005), no. 3, 399-420. MR**2152852 (2006b:60148)****7.**M. O. Androshchuk and Yu. S. Mishura,*An estimate of the ruin probability for an insurance company investing in a BS-market*, Ukrain. Matem. Zh.**59**(2007), no. 11, 1143-1153. (to appear) MR**2402185****8.**S. Asmussen,*Ruin Probabilities*, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR**1794582 (2001m:62119)****9.**P. E. Protter,*Stochastic Integration and Differential Equations*, Springer, Berlin, 2004. MR**2020294 (2005k:60008)****10.**D. K. Faddeev and I. S. Sominskiĭ,*Problems in Higher Algebra*, ``Nauka'', Moscow, 1972; English transl., W. H. Freeman and Co., San Francisco-London, 1965. MR**0176990 (31:1258)****11.**R. J. Elliott,*Stochastic Calculus and Applications*, Springer, New York, 1982. MR**678919 (85b:60059)****12.**T. A. Kiffe,*A Discontinuous Volterra Integral Equation*, J. Integral Equations**1**(1979), 193-200. MR**540826 (80i:45006)**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
60H05,
60G15

Retrieve articles in all journals with MSC (2000): 60H05, 60G15

Additional Information

**M. O. Androshchuk**

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:
andr_m@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-08-00726-6

Keywords:
Semimartingale,
local martingale,
ruin probability,
investment strategy

Received by editor(s):
July 28, 2006

Published electronically:
July 10, 2008

Article copyright:
© Copyright 2008
American Mathematical Society