On the Moments of the Time of Ruin with Applications to Phase-Type Claims

Abstract We describe an approach to the evaluation of the moments of the time of ruin in the classical Poisson risk model. The methodology employed involves the expression of these moments in terms of linear combinations of convolutions involving compound negative binomial distributions. We then adapt the results for use in the practically important case involving phase-type claim size distributions. We present numerical examples to illuminate the influence of claim size variability on the moments of the time of ruin.

[1]  Symbolic calculation of the moments of the time of ruin , 2004 .

[2]  M. Kendall,et al.  Kendall's advanced theory of statistics , 1995 .

[3]  Stuart A. Klugman,et al.  Loss Models: From Data to Decisions , 1998 .

[4]  G. Willmot,et al.  Equilibrium compound distributions and stop-loss moments , 2005 .

[5]  C. Segerdahl.  When does ruin occur in the collective theory of risk , 1955 .

[6]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[7]  Howard R. Waters,et al.  Loss Models: from Data to Decisions. By Stuart Klugman, Harry Panjer and Gordon Willmot [John Wiley & Sons, New York, 1998] , 1999 .

[8]  Tomasz Rolski,et al.  Lundberg Approximations for Compound Distributions With Insurance Applications , 2002 .

[9]  Gordon E. Willmot,et al.  On the Density and Moments of the Time of Ruin with Exponential Claims , 2003, ASTIN Bulletin.

[10]  N. Wikstad.  Exemplification of Ruin Probabilities , 1971 .

[11]  G. Willmot,et al.  The Density of the Time to Ruin in the Classical Poisson Risk Model , 2005, ASTIN Bulletin.

[12]  Shaun S. Wang,et al.  Exponential and scale mixtures and equilibrium distributions , 1998 .

[13]  G. Willmot.  On higher-order properties of compound geometric distributions , 2002, Journal of Applied Probability.

[14]  P. Davies,et al.  Kendall's Advanced Theory of Statistics. Volume 1. Distribution Theory , 1988 .

[15]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[16]  M. Kendall,et al.  Kendall's Advanced Theory of Statistics: Volume 1 Distribution Theory , 1987 .

[17]  Paul Embrechts,et al.  Stochastic processes in insurance and finance , 2001 .

[18]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[19]  R. Friedland.  Life Expectancy in the Future , 1998 .

[20]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[21]  Howard R. Waters,et al.  The Distribution of the time to Ruin in the Classical Risk Model , 2002, ASTIN Bulletin.