On generalized Piterbarg-Berman function

Abstract

This paper aims to evaluate the Piterbarg-Berman function given by $$\mathcal{P\!B}_\alpha^h(x, E) = \int_\mathbb{R}e^z\mathbb{P} \left\{{\int_E \mathbb{I}\left(\sqrt2B_\alpha(t) - |t|^\alpha - h(t) - z>0 \right) {\text{d}} t > x} \right\} {\text{d}} z,\quad x\in[0, {mes}(E)],$$ with $h$ a drift function and $B_\alpha$ a fractional Brownian motion (fBm) with Hurst index $\alpha/2\in(0,1]$, i.e., a mean zero Gaussian process with continuous sample paths and covariance function \begin{align*} {\mathrm{Cov}}(B_\alpha(s), B_\alpha(t)) = \frac12 (|s|^\alpha + |t|^\alpha - |s-t|^\alpha). \end{align*} This note specifies its explicit expression for the fBms with $\alpha=1$ and $2$ when the drift function $h(t)=ct^\alpha, c>0$ and $E=\mathbb{R}_+\cup\{0\}$. For the Gaussian distribution $B_2$, we investigate $\mathcal{P\!B}_2^h(x, E)$ with general drift functions $h(t)$ such that $h(t)+t^2$ being convex or concave, and finite interval $E=[a,b]$. Typical examples of $\mathcal{P\!B}_2^h(x, E)$ with $h(t)=c|t|^\lambda-t^2$ and several bounds of $\mathcal{P\!B}_\alpha^h(x, E)$ are discussed. Numerical studies are carried out to illustrate all the findings. Keywords: Piterbarg-Berman function; sojourn time; fractional Brownian motion; drift function