On generalized Piterbarg-Berman function

Abstract

This paper aims to evaluate the Piterbarg-Berman function given by $${\mathcal{P}\mathcal{B}}_{\hat{I}}{\pm }^h(x,E)={\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{{\int }_E\mathbb{I}\left(\sqrt{2}{B}_{\hat{I}}\pm (t)-|t{|}^{\hat{I}}\pm -h(t)-z>0\right)\text{d}t>x\right\}\text{d}z,x\in [0,mes(E)],$$ with $h$ a drift function and $B_{\hat{I}}\pm$ a fractional Brownian motion (fBm) with Hurst index $\hat{I}\pm /2\in (0,1]$, i.e., a mean zero Gaussian process with continuous sample paths and covariance function \begin{align*} {\mathrm{Cov}}(B_α(s), B_α(t)) = \frac12 (|s|^α+ |t|^α- |s-t|^α). \end{align*} This note specifies its explicit expression for the fBms with $\hat{I}\pm =1$ and $2$ when the drift function $h(t)=c{t}^{\hat{I}}\pm ,c>0$ and $E={\mathbb{R}}_{+}\cup \{0\}$. For the Gaussian distribution $B_2$, we investigate ${\mathcal{P}\mathcal{B}}_2^h(x,E)$ 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 ${\mathcal{P}\mathcal{B}}_2^h(x,E)$ with $h(t)=c|t{|}^{\hat{I}}»-t^2$ and several bounds of ${\mathcal{P}\mathcal{B}}_{\hat{I}}{\pm }^h(x,E)$ are discussed. Numerical studies are carried out to illustrate all the findings. Keywords: Piterbarg-Berman function; sojourn time; fractional Brownian motion; drift function