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

Consider a random walk or Lévy process {S_t} and let τ(u) = inf {t ≥ 0:S_t > u}, ℙ^(u)(·) = ℙ(· | τ(u) < ∞). 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 τ(u) is described as u → ∞. 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 τ(u)/a(u) is either Pareto or exponential, and corresponding approximations for the finite time ruin probabilities are given.