2011/06/16 by Philip S Griffin, Philip S. Griffin, Ross A Maller +5
Decision Sciences · Economics, Econometrics and Finance · Mathematics · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1106.3292
arxiv created 2011/06/16 · openalex publication_date 2011/06/16 · arxiv updated 2011/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recent models of the insurance risk process use a Lévy process to generalise the traditional Cramér-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Lévy process which drifts to -∞ and satisfies a Cramér or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramér case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cramér model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the Lévy process belongs to the "GTSC" class.