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Dependent Risk Models with Bivariate Phase-Type Distributions

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

References

Abou-Kandil, H., Freiling, G., Ionescu, V. and Jank, G. (2003). Matrix Riccati Equations. Birkhäuser, Basel.CrossRefGoogle Scholar
Ahn, S. and Ramaswami, V. (2005). Efficient algorithms for transient analysis of stochastic fluid flow models. J. Appl. Prob. 42, 531–549.Google Scholar
Ahn, S. and Ramaswami, V. (2006). Transient analysis of fluid flow models via elementary level-crossing arguments. Stoch. Models 22, 129–147.CrossRefGoogle Scholar
Ahn, S., Badescu, A. L. and Ramaswami, V. (2007). Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Systems 55, 207–222.Google Scholar
Albrecher, H. and Teugels, J. L. (2006). Exponential behavior in the presence of dependence in risk theory. J. Appl. Prob. 43, 257–273.Google Scholar
Asmussen, S., Avram, F. and Usabel, M. (2002). Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bull. 32, 267–281.Google Scholar
Assaf, D., Langberg, N. A., Savits, T. H. and Shaked, M. (1984). Multivariate phase-type distributions. Operat. Res. 32, 688–702.Google Scholar
Badescu, A., Drekic, S. and Landriault, D. (2007). Analysis of a threshold dividend strategy for a MAP risk model. Scand. Actuarial J. 2007, 227–247.Google Scholar
Badescu, A. et al. (2005). Risk processes analyzed as fluid queues. Scand. Actuarial J. 2005, 127–141.CrossRefGoogle Scholar
Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2005). Algorithms for return probabilities for stochastic fluid flows. Stoch. Models 21, 149–184.Google Scholar
Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006). On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuarial J. 2005, 265–285.Google Scholar
Cai, J. and Li, H. (2007). Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J. Multivariate Anal. 98, 757–773.Google Scholar
Cheung, E. C. K., Landriault, D., Willmot, G. E. and Woo, J.-K. (2009). Structural properties of Gerber–Shiu functions in dependent Sparre Andersen models. Submitted.Google Scholar
Cossette, H., Marceau, E. and Marri, F. (2009). On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula. Insurance Math. Econom. 43, 444–455.Google Scholar
Faddeev, D. K. and Faddeeva, V. N. (1963). Computational Methods of Linear Algebra. Freeman and Company, San Francisco, CA.Google Scholar
Freund, J. E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1997). The Joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math. Econom. 21, 129–137.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. North Amer. Actuarial J. 2, 48–78.Google Scholar
Guo, C.-H. (2001). Nonsymmetric algebraic Riccati equations and Wiener–Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl. 23, 225–242.Google Scholar
Kulkarni, V. G. (1989). A new class of multivariate phase type distributions. Operat. Res. 37, 151–158.CrossRefGoogle 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 Math. Econom. 42, 600–608.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2005). On a general class of risk processes: analysis of the Gerber–Shiu function. Adv. Appl. Prob. 37, 836–856.CrossRefGoogle Scholar
Marshall, A. L. and Olkin, I. (1967). A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.Google Scholar
Neuts, M. (1979). A versatile markovian point process. J. Appl. Prob. 16, 764–779.Google Scholar
Ramaswami, V. (2006). Passage times in fluid models with application to risk processes. Methodology Comput. Appl. Prob. 8, 497–515.CrossRefGoogle Scholar
Ren, J. (2007). The discounted Joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model. North Amer. Actuarial J. 11, 128–136.CrossRefGoogle Scholar
Willmot, G. E. (2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance Math. Econom. 41, 17–31.CrossRefGoogle Scholar