Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-11T17:04:12.478Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilities of ruin when the safety loading tends to zero

Part of:Mathematical economicsLimit theorems

Published online by Cambridge University Press:  01 July 2016

Vsevolod K. Malinovskii
Vsevolod K. Malinovskii*
Affiliation:
Steklov Mathematical Institute
*
∗ Postal address: Steklov Mathematical Institute, Gubkina str 8, 117966 Moscow GSP-1, Russia. Email address: malinov@mi.ras.ru
Get accessRights & Permissions [Opens in a new window]

Abstract

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero asu→∞.

Keywords

Ruin theoryAndersen's risk modelcapital-dependentsafety loadingsapproximations

MSC classification

Primary:91B30: Risk theory, insurance
Secondary:60F99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000, pp. 885 - 923
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C.Google Scholar
Adams, E. (1922). Smithsonian Mathematical Formulae. Washington.Google Scholar
Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuar. J. 67, 31–57; ibid. (1985), 64.Google Scholar
Barndorff-Nielsen, O. and Schmidli, H. (1995). Saddlepoint approximations for the probability of ruin in finite time Scand. Actuar. J. 169–186.Google Scholar
Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley, New York.Google Scholar
Billingsley, P. (1979). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bolthausen, E. (1980). The Berry–Esseen theorem for functionals of discrete Markov chains. Z. Wahrscheinlichkeitsth. 54, 59–73.CrossRefGoogle Scholar
Copson, E. T. (1965). Asymptotic Expansions. Cambridge University Press.Google Scholar
Cramér, H., (1930). On the Mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm. Reprinted in Harald Cramér Collective Works, Vol. 1 (1994). ed. A. Martin-Löf. Springer, Berlin, pp. 601–678.Google Scholar
Cramér, H., (1969). Historical review of Filip Lundberg's works on risk theory. Skand. Act. Tidskr. 52, 6–12; Suppl.Google Scholar
Dubinskaite, I. (1984). Limit theorems in Rk. II–III. Lithuanian Math. J. 24, 256–265, 352–334.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. 2nd edn., vol. 2. John Wiley, New York.Google Scholar
Lundberg, F. (1926). Försäkringsteknisk Riskutjämning. F. Englunds, Stockholm.Google Scholar
Malinovskii, V. K. (1993). Limit theorems for stopped random sequences I: Rates of convergence and asymptotic expansions. Theory Prob. Appl. 38, 673–693.Google Scholar
Malinovskii, V. K. (1994). Corrected normal approximation for the probability of ruin within finite time. Scand. Actuar. J. 161–174.Google Scholar
Malinovskii, V. K. (1996). Approximations and upper bounds on probabilities of large deviations in the problem of ruin within finite time. Scand. Actuar. J. 124–147.Google Scholar
Malinovskii, V. K. (1998a). Non-poissonian claims arrivals and calculation of the probability of ruin. Insurance: Math. Econ. 22, 123–138.Google Scholar
Malinovskii, V. K. (1998b). Some aspects of rate making and collective risk models with variable safety loadings. Trans. 26th Int. Congr. Actuaries 4, 465–481.Google Scholar
Malinovskii, V. K. (1998c). Probabilities of ruin when the safety loading tends to zero. Working Paper No. 153, Laboratory of Actuarial Mathematics. University of Copenhagen.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.Google Scholar
Petrov, V. V. (1987). Limit Theorems for Sums of Independent Random Variables. Na-uka, Moscow (in Russian).Google Scholar
Saulis, L. I. (1969). Asymptotic expansions of probabilities of large deviations. Litovskii Matemat. Sbornik (Lietuvos Matematikos Rinkinys) 9, 605–625 (in Russian).Google Scholar
Siegmund, D. (1985). Sequential Analysis. Tests and Confidence Intervals. Springer, New York.Google Scholar
Smith, W. L. (1962). On some general renewal theorems for nonidentically distributed variables. Proc. Fourth Berkeley Symp. Math. Stat. and Prob., Vol. 2. University of California Press, Berkeley, 467–514.Google Scholar
Titchmarsh, E. C. (1951). The Theory of the Riemann Zeta-function. Oxford University Press.Google Scholar
Tricomi, F. G. and Erdélyi, A. (1951). The asymptotic expansions of a ratio of Gamma functions. Pacific J. Math. 1, 133–142.Google Scholar
Von Bahr, B. (1974). Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuar. J. 57, 190–204.Google Scholar
Whittaker, E. T. and Watson, G. N. (1963). A Course of Modern Analysis. 4th edn. Cambridge University Press.Google Scholar
Wishart, J. (1957). An approximate formula for the cumulative z-distribution. Ann. Math. Statist. 28, 504–510.Google Scholar
Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.Google Scholar