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First-crossing and ballot-type results for some nonstationary sequences

Part of: Stochastic processes Applications Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Claude Lefèvre
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium. Email address: clefevre@ulb.ac.be
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Abstract

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In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

Keywords

Parametric distributionsstability by convolutionfirst-crossing of partial sumsgeneralized Abel-Gontcharoff polynomialsdiscrete distributionstotal progenybusy periodruin probability

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60E05: Distributions 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 492 - 509
Copyright
Copyright © Applied Probability Trust 2007 

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