Perturbed Risk Processes Analyzed as Fluid Flows

In this article, we present a unified phase-type approach for calculating ruin probabilities for a class of perturbed risk processes, which includes the perturbed Sparre–Andersen process with phase-type interclaim time and claim size distributions, the perturbed Markov-modulated risk process, and the MAP/PH risk process as well. The key to the solution method is the identification for a given risk process of a parallel Markov-modulated fluid flow. The resulting group of perturbed risk processes amenable to this approach we refer to as the Markov-modulated fluid-flow equivalent (MFE) class, as the ability to find an equivalent fluid flow is the only limiting factor in our analysis. The primary contributions of this work are 1) the unified, tractable, phase-type structures for the maximal aggregate loss and ladder height distributions for all variants of perturbed risk processes belonging to the MFE class, and 2) the explicit formulas for the ruin probability for all variants in this class. The full breadth of the models is illustrated through a diverse range of examples.

[1]  José Garrido,et al.  The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion , 2005 .

[2]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[3]  R. Wolpert.  Lévy Processes , 2000 .

[4]  Martin Jacobsen,et al.  The time to ruin for a class of Markov additive risk process with two-sided jumps , 2005, Advances in Applied Probability.

[5]  Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion , 1994 .

[6]  Guy Latouche,et al.  Risk processes analyzed as fluid queues , 2005 .

[7]  Olof Thorin,et al.  Numerical evaluation of ruin probabilities for a finite period , 1973, ASTIN Bulletin.

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

[9]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[10]  Yi Lu,et al.  The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion , 2007 .

[11]  Cramér-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion , 1995 .

[12]  E. Frostig.  On Ruin Probability for a Risk Process Perturbed by a Lévy Process with no Negative Jumps , 2008 .

[13]  David A. Stanford,et al.  Erlangian approximation to finite time ruin probabilities in perturbed risk models , 2007, Numerical Methods for Structured Markov Chains.

[14]  Hailiang Yang,et al.  On Erlang(2) Risk Process Perturbed by Diffusion , 2005 .

[15]  Olof Thorin,et al.  Calculation of Ruin Probabilities when the Claim Distribution is Lognormal , 1977, ASTIN Bulletin.

[16]  D. Dickson,et al.  Lundberg Approximations for Compound Distributions with Insurance Applications . By G. E. Willmot and X. S. Lin. (Springer, 2000) , 2001, British Actuarial Journal.

[17]  S. Asmussen.  Stationary distributions for fluid flow models with or without Brownian noise , 1995 .

[18]  Lothar Breuer,et al.  The use of vector-valued martingales in risk theory , 2008 .

[19]  Gordon E. Willmot,et al.  Lundberg approximations for compound distributions with insurance applications , 2001 .