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

Journal article, 2004

Let $X = \{X(t),\ t \ge 0\}$ be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index $H > 0$. For given constants $c > 0$ and $\gamma > H$, consider the storage process $$Y(t) = \sup_{s \ge t}\big(X(s) - X(t) - c(s-t)^\gamma\big)\quad \text{for} t \ge 0.$$ There has been considerable interest in recent years in the study of the behavior of the following probability of overload during the time period $[0, t]$: $${\bf P}\left\{\sup_{s \in [0, t]} Y(s) > u\right\}\quad \text{as}\ u \to \infty.$$