Large deviations results for subexponential tails, with applications to insurance risk

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:64

Article

Asmussen, S ; Kluppelberg, C

关键词: conditioned limit theorem;    downwards skip-free process;    excursion;    extreme value theory;    insurance risk;    integrated tail;    maximum domain of attraction;    path decomposition;    random walk;    regular variation;    ruin probability;    subexponential distribution;    total variation convergence;   

DOI  :  10.1016/S0304-4149(96)00087-7

来源: Elsevier

【 摘 要 】

Consider a random walk or Levy process {S-t} and let tau(u) = inf{t greater than or equal to 0:S-t > u}, P-(u) (.) = P(.\tau(u) < infinity). Assuming that the upwards jumps are heavy-tailed, say subexponential (e.g. Pareto, Weibull or lognormal), the asymptotic form of the P-(u)-distribution of the process {S-t} up to time tau(u) is described as u --> infinity. Essentially, the results confirm the folklore that level crossing occurs as result of one big jump. Particular sharp conclusions are obtained for downwards skip-free processes like the classical compound Poisson insurance risk process where the formulation is in terms of total variation convergence. The ideas of the proof involve excursions and path decompositions for Markov processes. As a corollary, it follows that for some deterministic function a(u), the limiting P-(u)-distribution of tau(u)/a(u) is either Pareto or exponential, and corresponding approximations for the finite time ruin probabilities are given.

【 授权许可】

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