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

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 .>H and c>0. Let . be the Lévy measure on .[0,.) of X, and suppose that R(u)..({y..[0,.):sup.t.0y(t)/(1+ct.)>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).), we show that P{sup.s.[0,t(u)]Y(s)>u}.P{Y({t.}(u))>u} as u.., when 0.t.(u).t(u) do not grow too fast with u [e.g., t(u)=o(u1/.)].