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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
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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

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