Large deviations of a storage process with fractional Brownian motion as input
A storage process \(Y(t)=\sup_{\sigma>t}(X(\sigma)-X(t)-c(t-\sigma))\) is considered, where \(X(t)\) is a fractional Brownian motion with the Hurst parameter \(H\), \(0<H<1\). Asymptotic of the probability \(p(u)=\Pr\{\sup_{t\in[0,T]}Y(t)>u\}\) is investigated for a fixed \(T\) and large \(u\). E.g., if \(0<H<1/2\), then \[ p(u)=C u^{-1}(Au^{1-H})^{2/H-1}\Psi(Au^{1-H})(1+o(1)) \] as \(u\to\infty\), where \(A=(1-H)^{-1}(H/(c(1-H)))^{-H}\), \(\Psi(x)=\exp(-x^2/2)/(\sqrt{2\pi}x)\).
scientific article; zbMATH DE number 1833863