Tang, Qihe and Tsitsiashvili, Gurami (2004) Finite and Infinite Time Ruin Probabilities in the Presence of Stochastic Returns on Investments. Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.
- Published Version
This paper investigates the finite and infinite time ruin probabilities in a discrete time stochastic economic environment. Under the assumption that the insurance risk
- the total net loss within one time period - is extended-regularly-varying or rapidly varying tailed, various precise estimates for the ruin probabilities are derived. In
particular, some estimates obtained are uniform with respect to the time horizon, hence apply for the case of infinite time ruin.
|Divisions:||Concordia University > Faculty of Arts and Science > Mathematics and Statistics|
|Item Type:||Monograph (Technical Report)|
|Authors:||Tang, Qihe and Tsitsiashvili, Gurami|
|Series Name:||Department of Mathematics & Statistics. Technical Report No. 14/04|
|Corporate Authors:||Concordia University. Department of Mathematics & Statistics|
|Keywords:||Asymptotics; class S(√); endpoint; extended regular variation; financial risk; insurance risk; rapid variation; ruin probability|
|Deposited By:||DIANE MICHAUD|
|Deposited On:||02 Jun 2010 16:29|
|Last Modified:||08 Dec 2010 23:24|
Asmussen, S. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 (1998), no. 2, 354-374.
Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Cambridge University Press, Cambridge, 1987.
Brandt, A. The stochastic equation Yn+1 = AnYn + Bn with stationary coefficients. Adv. in Appl. Probab. 18 (1986), no. 1, 211-220.
Breiman, L. On some limit theorems similar to the arc-sin law. (Russian) Teor. Verojatnost. i Primenen. 10 (1965), 351-360; translation in Theor. Probability Appl. 10(1965), 323-331.
Cai, J.; Tang, Q. Lp transform and asymptotic ruin probability in a perturbed risk model. Stochastic Process. Appl. (2004), to appear.
Chistyakov, V. P. A theorem on sums of independent positive random variables and its applications to branching random processes. (Russian) Teor. Verojatnost. i Primenen 9 (1964), 710-718; translation in Theory Prob. Appl. 9 (1964), 640-648.
Chover, J.; Ney, P.; Wainger, S. Functions of probability measures. J. Analyse Math. 26 (1973a), 255-302.
Chover, J.; Ney, P.; Wainger, S. Degeneracy properties of subcritical branching processes. Ann. Probability 1 (1973b), 663-673.
Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Process. Appl. 49 (1994), no. 1, 75-98.
Davis, R.; Resnick, S. Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stochastic Process. Appl. 30 (1988), no. 1, 41-68.
Embrechts, P. A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Probab. 20 (1983), no. 3, 537-544.
Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling extremal events for insurance and finance. Springer-Verlag, Berlin, 1997.
Frolova, A.; Kabanov, Y.; Pergamenshchikov, S. In the insurance business risky investments are dangerous. Finance Stoch. 6 (2002), no. 2, 227-235.
Geluk, J. L.; de Haan, L. Regular variation, extensions and Tauberian theorems. Amsterdam, 1987.
Goldie, C. M. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991), no. 1, 126-166.
de Haan, L. On regular variation and its application to the weak convergence of sample extremes. Amsterdam, 1970.
Kalashnikov, V.; Konstantinides, D. Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27 (2000), no. 1, 145-149.
Kalashnikov, V.; Norberg, R. Power tailed ruin probabilities in the presence of risky investments. Stochastic Process. Appl. 98 (2002), no. 2, 211-228.
Klüppelberg, C. Subexponential distributions and characterizations of related classes. Probab. Theory Related Fields 82 (1989), no. 2, 259-269.
Klüppelberg, C.; Mikosch, T. Large deviations of heavy-tailed random sums with applications in insurance and finance. J. Appl. Probab. 34 (1997), no. 2, 293-308.
Klüppelberg, C.; Stadtmüller, U. Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuar. J. (1998), no. 1, 49-58.
Konstantinides, D.; Tang, Q.; Tsitsiashvili, G. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31 (2002), no. 3, 447-460.
Ng, K. W.; Tang, Q.; Yan, J.; Yang, H. Precise large deviations for the prospective-loss process. J. Appl. Probab. 40 (2003), no. 2, 391-400.
Norberg, R. Ruin problems with assets and liabilities of diffusion type. Stochastic Process. Appl. 81 (1999), no. 2, 255-269.
Nyrhinen, H. On the ruin probabilities in a general economic environment. Stochastic Process. Appl. 83 (1999), no. 2, 319-330.
Nyrhinen, H. Finite and infinite time ruin probabilities in a stochastic economic environment. Stochastic Process. Appl. 92 (2001), no. 2, 265-285.
Paulsen, J. On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Probab. 12 (2002), no. 4, 1247-1260.
Resnick, S. I. Extreme values, regular variation, and point processes. Springer-Verlag,
New York, 1987.
Rogozin, B. A. On the constant in the definition of subexponential distributions. (Russian) Teor. Veroyatnost. i Primenen. 44 (1999), no. 2, 455-458; translation in Theory Probab. Appl. 44 (2000), no. 2, 409-412.
Rogozin, B. A.; Sgibnev, M. S. Banach algebras of measures on the line with given asymptotics of distributions at infinity. (Russian) Sibirsk. Mat. Zh. 40 (1999), no. 3, 660-672; translation in Siberian Math. J. 40 (1999), no. 3, 565-576.
Sgibnev, M. S. On the distribution of the maxima of partial sums. Statist. Probab. Lett. 28 (1996), no. 3, 235-238.
Sundt, B.; Teugels, J. L. Ruin estimates under interest force. Insurance Math. Econom. 16 (1995), no. 1, 7-22.
Sundt, B.; Teugels, J. L. The adjustment function in ruin estimates under interest force. Insurance Math. Econom. 19 (1997), no. 2, 85-94.
Tang, Q. The ruin probability of a discrete time risk model under constant interest rate with heavy tails. Scand. Actuar. J. (2004), no. 3, 229-240.
Tang, Q.; Tsitsiashvili, G. Precise estimates for the ruin probability infinite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process. Appl. 108 (2003), no. 2, 299-325.
Teugels, J. L. The class of subexponential distributions. Ann. Probability 3 (1975), no.6, 1000-1011.
Tsitsiashvili, G. Quality properties of risk models under stochastic interest force. Inform. Process. 2 (2002), no. 2, 264-268.
Vervaat, W. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 (1979), no. 4, 750-783.
Yang, H. Non-exponential bounds for ruin probability with interest effect included. Scand. Actuar. J. (1999), no. 1, 66-79.
Repository Staff Only: item control page
Downloads per month over past year