×

On the supremum of \(\gamma\)-reflected processes with fractional Brownian motion as input. (English) Zbl 1316.60054

Summary: Let \(\{X_{H}(t): t \geq 0\}\) be a fractional Brownian motion with Hurst index \(H \in (0,1]\) and define a \(\gamma\)-reflected process \(W_{\gamma}(t)=X_{H}(t)-ct-\gamma \inf_{s \in [0,t]}(X_{H}(s)-cs)\), \(t \geq 0\), with \(c>0\), \(\gamma \in [0,1]\) two given constants. In this paper, we establish the exact tail asymptotic behaviour of \(M_{\gamma}(T)= \sup_{t \in[0,T]}W_{\gamma}(t)\) for any \(T \in (0,\infty]\). Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G70 Extreme value theory; extremal stochastic processes
60G15 Gaussian processes
60G60 Random fields
PDFBibTeX XMLCite
Full Text: DOI arXiv