Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T14:39:23.681Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTRIBUTION OF THE TIME TO RUIN IN SOME SPARRE ANDERSEN RISK MODELS

Published online by Cambridge University Press:  29 April 2013

Tianxiang Shi  and
David Landriault
Tianxiang Shi
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Canada E-mail: [email protected]
David Landriault
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely the combination of n exponentials. A remark is further made to emphasize that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined.

Keywords

Time to ruinfinite-time ruin probabilitymultivariate Lagrange expansion theoremduration of a busy periodcombination of n exponentialsKm/G/1 queueing modelnumber of claims until ruin
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 43 , Issue 1 , January 2013 , pp. 39 - 59
Copyright
Copyright © ASTIN Bulletin 2013

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.)

References

Asmussen, S. (1995) Stationary distributions for fluid flow models with or without Brownian noise. Communications in Statistics. Stochastic Models, 11 (1), 21–49.CrossRefGoogle Scholar
Avanzi, B., Gerber, H.U. and Shiu, E.S.W. (2007) Optimal dividends in the dual model. Insurance: Mathematics and Economics, 41, 111–123.Google Scholar
Borovkov, K.A. and Dickson, D.C.M. (2008) On the ruin time distribution for a Sparre Andersen risk process with exponential claim sizes. Insurance: Mathematics and Economics, 42, 1104–1108.Google Scholar
Cohen, J.W. (1982) The Single Server Queue, 2nd ed.Amsterdam: North-Holland.Google Scholar
Dickson, D.C.M. and Li, S. (2010) Finite time ruin problems for the Erlang(2) risk model. Insurance: Mathematics and Economics, 46 (1), 12–18.Google Scholar
Dickson, D. and Li, S. (2012) Erlang Risk Models and Finite Time Ruin Problems. Scandinavian Actuarial Journal, 3, 183–202.CrossRefGoogle Scholar
Dickson, D.C.M. and Willmot, G.E. (2005) The density of the time to ruin in the classical Poisson risk model. ASTIN Bulletin, 35 (1), 45–60.CrossRefGoogle Scholar
Drekic, S. and Willmot, G.E. (2003) On the density and moments of the time of ruin with exponential claims. ASTIN Bulletin, 33, 11–21.CrossRefGoogle Scholar
Dufresne, D. (2007) Fitting combinations of exponentials to probability distributions. Applied Stochastic Models in Business and Industry, 23 (1), 23–48.CrossRefGoogle Scholar
Frostig, E. (2004) Upper bounds on the expected time to ruin and on the expected recovery time. Advances in Applied Probability, 36, 377–397.CrossRefGoogle Scholar
Garcia, J.M.A. (2005) Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin, 35 (1), 113–130.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (2005) The time value of ruin in a Sparre Andersen model. North American Actuarial Journal, 9 (2), 49–69.CrossRefGoogle Scholar
Good, I.J. (1960) Generalizations to several variables of Lagrange expansion, with applications to stochastic processes. Proceedings of the Cambridge Philosophical Society, 56, 367–380.CrossRefGoogle Scholar
Goulden, I.P. and Jackson, D.M. (1983) Combinatorial Enumeration. New York: Wiley.Google Scholar
Khan, I.R. and Ohba, R. (2003) Taylor series based finite difference approximations of higher-degree derivatives. Journal of Computational and Applied Mathematics, 154 (1), 115–124.CrossRefGoogle Scholar
Kleinrock, L. (1975) Queueing Systems. Vol. I: Theory. New York: Wiley.Google Scholar
Landriault, D., Shi, T. and Willmot, G.E. (2011) Joint densities involving the time to ruin in the Sparre–Andersen risk model with exponential claim sizes. Insurance: Mathematics and Economics, 49, 371–379.Google Scholar
Landriault, D. and Willmot, G.E. (2008) On the Gerber–Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. Insurance: Mathematics and Economics, 42, 600–608.Google Scholar
Li, S. and Garrido, J. (2005) On a general class of risk processes: Analysis of the Gerber–Shiu function. Advances in Applied Probability, 37 (3), 836–856.CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (1997) The probability of ruin in finite time with discrete claim size distribution. Scandinavian Actuarial Journal, 1, 58–69.CrossRefGoogle Scholar
Poincaré, H. (1886) Sur les résidus des intégrales doubles. Acta Mathematics, 9, 321–380.CrossRefGoogle Scholar
Ramaswami, V. (2006) Passage times in fluid models with application to risk processes. Methodology and Computing in Applied Probability, 8 (4), 497–515.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. New York: Wiley.Google Scholar
Willmot, G.E. and Woo, J.K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–115.CrossRefGoogle Scholar
Willmot, Gordon E. and Lin, X. Sheldon. (2011) Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27 (1), 2–16.CrossRefGoogle Scholar