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Applications of fluid flow matrix analytic methods in ruin theory – a review. (English) Zbl 1186.60092
From the introduction: “The main focus of this review paper is to present a recent area of applications of matrix analytic methods (MAMs), namely ruin theory. By making use of the connection between an insurers surplus process and a particular fluid queue, it presents a unified methodology for studying a large class of insurance risk models via the recent developments in MAMs for the analysis of fluid queues. MAMs are based on matrix calculations that often have nice probabilistic interpretations. The forms of these quantities provide algorithmic tractability, a feature that most of the traditional analytic techniques do not possess. MAMs make no use of the theory of eigenvalues. This is in total accordance with the fact that the fundamental matrices have probabilistic meanings, as the eigenvalues do not. The applications of MAMs in ruin theory provide the derivation of an expression for certain ruin-related quantities without directly involving the roots of the so-called Lundberg fundamental equation. The role of the Lundberg equation roots in the traditional analytic approach is replaced in MAMs by a matrix quantity representing the Laplace-Stieltjes transform of a busy period in a particular fluid queue.”

60K15 Markov renewal processes, semi-Markov processes
91B30 Risk theory, insurance (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
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