On overload in a storage model, with a self-similar and infinitely divisible input

Abstract

Let {X(t)}_t≥0 be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index H>0. Pick constants γ>H and c>0. Let ν be the Lévy measure on ℝ^[0,∞) of X, and suppose that R(u)≡ν({y∈ℝ^[0,∞):sup _t≥0y(t)/(1+ct^γ)>u}) is suitably “heavy tailed” as u→∞ (e.g., subexponential with positive decrease). For the “storage process” Y(t)≡sup _s≥t(X(s)−X(t)−c(s−t)^γ), we show that P{sup _s∈[0,t(u)]Y(s)>u}∼P{Y({t̂}(u))>u} as u→∞, when 0≤t̂(u)≤t(u) do not grow too fast with u [e.g., t(u)=o(u^1/γ)].