Power estimates for ruin probabilities
Harri Nyrhinen
Source: Adv. in Appl. Probab. Volume 37, Number 3
(2005), 726-742.
Abstract
Let X1, X2,... be real-valued random variables. For u>0, define the time of ruin T = T(u) by T = inf{n: X1+⋯+Xn>u} or T=∞ if X1+⋯+Xn≤u for every n = 1,2,.... We are interested in the ruin probabilities of general processes {Xn} for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u-R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {Xn}. We also extend our results to the case of the infinite time horizon.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1127483744
Digital Object Identifier: doi:10.1239/aap/1127483744
Mathematical Reviews number (MathSciNet): MR2156557
Zentralblatt MATH identifier: 1087.60038
References
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354--374.
Mathematical Reviews (MathSciNet): MR1624933
Digital Object Identifier: doi:10.1214/aoap/1028903531
Project Euclid: euclid.aoap/1028903531
Zentralblatt MATH: 0942.60034
Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103--125.
Mathematical Reviews (MathSciNet): MR1419495
Digital Object Identifier: doi:10.1016/S0304-4149(96)00087-7
Zentralblatt MATH: 0879.60020
Asmussen, S., Fløe Henriksen, L. and Klüppelberg, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Process. Appl. 54, 29--43.
Mathematical Reviews (MathSciNet): MR1302693
Digital Object Identifier: doi:10.1016/0304-4149(93)00003-X
Zentralblatt MATH: 0814.60067
Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk processes with subexponential jumps. Adv. Appl. Prob. 31, 422--447.
Mathematical Reviews (MathSciNet): MR1724561
Zentralblatt MATH: 0942.60033
Digital Object Identifier: doi:10.1239/aap/1029955142
Project Euclid: euclid.aap/1029955142
Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117--130.
Mathematical Reviews (MathSciNet): MR2036276
Digital Object Identifier: doi:10.1239/jap/1077134672
Collamore, J. F. (1998). First passage times of general sequences of random vectors: a large deviations approach. Stoch. Process. Appl. 78, 97--130.
Mathematical Reviews (MathSciNet): MR1653300
Digital Object Identifier: doi:10.1016/S0304-4149(98)00056-8
Zentralblatt MATH: 0934.60025
Daykin, C. D., Pentikäinen, T. and Pesonen, M. (1994). Practical Risk Theory for Actuaries (Monogr. Statist. Appl. Prob. 53). Chapman and Hall, London.
Mathematical Reviews (MathSciNet): MR1284039
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.
Mathematical Reviews (MathSciNet): MR1619036
Zentralblatt MATH: 0896.60013
Duffy, K., Lewis, J. T. and Sullivan, W. G. (2003). Logarithmic asymptotics for the supremum of a stochastic process. Ann. Appl. Prob. 13, 430--445.
Mathematical Reviews (MathSciNet): MR1970270
Digital Object Identifier: doi:10.1214/aoap/1050689587
Project Euclid: euclid.aoap/1050689587
Zentralblatt MATH: 1032.60025
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55--72.
Mathematical Reviews (MathSciNet): MR652832
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1458613
Zentralblatt MATH: 0873.62116
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.
Mathematical Reviews (MathSciNet): MR270403
Fuc, D. H. and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Prob. Appl. 16, 643--660.
Hu, Y. (2004). Large deviations for generalized compound Poisson risk models and its bankruptcy moments. Sci. China Ser. A 47, 311--319.
Mathematical Reviews (MathSciNet): MR2068948
Digital Object Identifier: doi:10.1360/02ys0355
Zentralblatt MATH: 1065.60024
Hu, Y. and Nyrhinen, H. (2004). Large deviations viewpoints for heavy-tailed random walks. J. Theoret. Prob. 17, 761--768.
Mathematical Reviews (MathSciNet): MR2091560
Digital Object Identifier: doi:10.1023/B:JOTP.0000040298.43712.e8
Zentralblatt MATH: 1068.60038
Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998, 49--58.
Mathematical Reviews (MathSciNet): MR1626664
Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025--1064.
Mathematical Reviews (MathSciNet): MR1789987
Digital Object Identifier: doi:10.1214/aoap/1019487517
Project Euclid: euclid.aoap/1019487517
Zentralblatt MATH: 1083.60506
Nyrhinen, H. (1995). On the typical level crossing time and path. Stoch. Process. Appl. 58, 121--137.
Mathematical Reviews (MathSciNet): MR1341558
Digital Object Identifier: doi:10.1016/0304-4149(94)00079-9
Zentralblatt MATH: 0829.60023
Nyrhinen, H. (1998). Rough descriptions of ruin for a general class of surplus processes. Adv. Appl. Prob. 30, 1008--1026.
Mathematical Reviews (MathSciNet): MR1671093
Digital Object Identifier: doi:10.1239/aap/1035228205
Zentralblatt MATH: 0932.60046
Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319--330.
Mathematical Reviews (MathSciNet): MR1708212
Digital Object Identifier: doi:10.1016/S0304-4149(99)00030-7
Zentralblatt MATH: 0997.60041
Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265--285.
Mathematical Reviews (MathSciNet): MR1817589
Digital Object Identifier: doi:10.1016/S0304-4149(00)00083-1
Zentralblatt MATH: 1047.60040
Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Prob. 12, 1247--1260.
Mathematical Reviews (MathSciNet): MR1936592
Digital Object Identifier: doi:10.1214/aoap/1037125862
Project Euclid: euclid.aoap/1037125862
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.
Mathematical Reviews (MathSciNet): MR1680267
Zentralblatt MATH: 0940.60005
Tang, Q. and Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299--325.
Mathematical Reviews (MathSciNet): MR2019056
Digital Object Identifier: doi:10.1016/j.spa.2003.07.001
Zentralblatt MATH: 1075.91563
Tang, Q. and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob. 36, 1278--1299.
Mathematical Reviews (MathSciNet): MR2119864
Digital Object Identifier: doi:10.1239/aap/1103662967
Zentralblatt MATH: 1095.91040
Advances in Applied Probability
