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Dependent Risk Models with Bivariate Phase-Type Distributions

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Andrei L. Badescu ,
Eric C. K. Cheung  and
David Landriault
Andrei L. Badescu*
Affiliation:
University of Toronto
Eric C. K. Cheung*
Affiliation:
University of Waterloo
David Landriault*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Ontario, M5S 3G3, Canada.
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada.
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada.
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Abstract

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In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

Keywords

Surplus processbivariate phase-type distributionfluid queueGerber–Shiu functiondeficit at ruinsurplus prior to ruin

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 113 - 131
Copyright
Copyright © Applied Probability Trust 2009 

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