On generalized Piterbarg-Berman function

TL;DR

This paper derives explicit formulas and bounds for the generalized Piterbarg-Berman function involving fractional Brownian motions with specific drift functions, providing numerical insights into its behavior for different parameters.

Contribution

It offers explicit expressions for the Piterbarg-Berman function for certain fractional Brownian motions and drift functions, extending previous understanding and providing bounds and numerical illustrations.

Findings

Explicit formulas for Bm with lpha=1,2 and specific drifts.

Bounds for Bm with general drift functions and finite intervals.

Numerical results illustrating the behavior of the function under various conditions.

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

Equations

$${\mathcal{P}\mathcal{B}}_{\alpha }^h(x,E)={\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{{\int }_E\mathbb{I}\left(\sqrt{2}{B}_{\alpha }(t)-|t{|}^{\alpha }-h(t)-z>0\right)\text{d}t>x\right\}\text{d}z,x\in [0,mes(E)],$$

$$\mathrm{Cov}(B_{\alpha }(s),B_{\alpha }(t))=\frac{1}{2}(|s{|}^{\alpha }+|t{|}^{\alpha }-|s-t{|}^{\alpha }).$$

$$L_{u,T}:={\int }_0^T\mathbb{I}\left(X\left(t\right)>u\right)\text{d}t$$

$$p_{u,T}(x)=\mathbb{P}\left\{\nu (u)L_{u,T}>x\right\}\sim C(x)\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }X(t)>u\right\}$$

$$G(x)=\mathbb{P}\left\{{\int }_{\mathbb{R}}\mathbb{I}(\sqrt{2}{B}_{\alpha }(t)-|t{|}^{\alpha }+\mathcal{W}>0)\text{d}t>x\right\},x\ge 0,$$

$$\mathrm{Cov}(B_{\alpha }(s),B_{\alpha }(t))=\frac{1}{2}(|s{|}^{\alpha }+|t{|}^{\alpha }-|s-t{|}^{\alpha }).$$

$${\mathcal{P}\mathcal{B}}_{\alpha }^h(x,E):={\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{{\int }_E\mathbb{I}(\sqrt{2}{B}_{\alpha }(t)-|t{|}^{\alpha }-h(t)-z>0)\text{d}t>x\right\}\text{d}z,x\in [0,\mathrm{mes}(E)]$$

$${\mathcal{P}\mathcal{B}}_{\alpha }^h(x):=\underset{T\to \infty }{\lim }\frac{{\mathcal{P}\mathcal{B}}_{\alpha }^h(x,[0,T])}{T^{\mathbb{I}\left(h=0\right)}},x\ge 0$$

$${\mathcal{P}\mathcal{B}}_1^h(x)$$

$$-(1+c)e^{-cx}\left[\frac{1-c}{c}\Psi (x_c^{\prime })-x_c^{\prime }\phi (x_c^{\prime })+\frac{1+x_c^{\prime 2}/2}{2}{e}^{-x_c^{\prime 2}/2}\right]$$

$${\mathcal{P}\mathcal{B}}_2^h(x)=\sqrt{\frac{1+c}{c}}\Psi \left(\sqrt{\frac{c(1+c)}{2}}x\right)e^{-\frac{(1+c)x^2}{4}}+\Psi \left(\frac{1-c}{\sqrt{2}}x\right)e^{-c{x}^2}-\Psi \left(\frac{1+c}{\sqrt{2}}x\right),x\ge 0.$$

$${\tau }_z=\inf \{t\ge 0:\sqrt{2}{B}_1(t)-t-h(t)\ge z\},Y_z={\int }_0^{\infty }\mathbb{I}(\sqrt{2}{B}_1(t)-t-h(t)-z>0)\text{d}t,$$

$$f_{\mathcal{N}}(t)=h(t)+t^2-\sqrt{2}\mathcal{N}t+z,t\in \mathbb{R}$$

$$f_{\mathcal{N}}(s^{\ast })=\underset{s\in E}{\min }{f}_{\mathcal{N}}(s)<0,f_{\mathcal{N}}(t^{\ast })=\underset{t\in \mathbb{R}}{\min }{f}_{\mathcal{N}}(t)<0.$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)={\int }_{\mathbb{R}}{e}^z\left[\mathbb{P}\left\{f_{\mathcal{N}}(a)<0,f_{\mathcal{N}}(b)<0\right\}\text{d}z+\mathbb{P}\left\{f_{\mathcal{N}}(a)>0,f_{\mathcal{N}}(b)>0,s_2-s_1>x,f_{\mathcal{N}}(s^{\ast })<0\right\}\right]\text{d}z$$

$$+{\int }_{\mathbb{R}}{e}^z\left[\mathbb{P}\left\{f_{\mathcal{N}}(a)\le 0,f_{\mathcal{N}}(a+x)<0,f_{\mathcal{N}}(b)>0\right\}+\mathbb{P}\left\{f_{\mathcal{N}}(a)>0,f_{\mathcal{N}}(b-x)<0,f_{\mathcal{N}}(b)\le 0\right\}\right]\text{d}z.$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)={\int }_{\mathbb{R}}{e}^z\left[\mathbb{P}\left\{f_{\mathcal{N}}(a+x)<0,\sqrt{2}\mathcal{N}\le h_{+}^{\prime }(a)+2a\right\}+\mathbb{P}\left\{f_{\mathcal{N}}(b-x)<0,\sqrt{2}\mathcal{N}\ge h_{-}^{\prime }(b)+2b\right\}\right]\text{d}z$$

$$+{\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{\min (b,t_2)-\max (a,t_1)>x,h_{+}^{\prime }(a)+2a<\sqrt{2}\mathcal{N}<h_{-}^{\prime }(b)+2b,f_{\mathcal{N}}(t^{\ast })<0\right\}\text{d}z.$$

$${\mathcal{P}\mathcal{B}}_2^h(x,[a,b])={\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{{\mathcal{L}}_{\mathcal{N}}(z)>x\right\}\text{d}z={\mathcal{P}\mathcal{B}}_2^h(x,[-b,-a]),x\in [0,b-a].$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)={\int }_{\mathbb{R}}{e}^z\left[\mathbb{P}\left\{f_{\mathcal{N}}(a+x)<0,\sqrt{2}\mathcal{N}/c<2a+x\right\}+\mathbb{P}\left\{f_{\mathcal{N}}(b-x)<0,\sqrt{2}\mathcal{N}/c>2b-x\right\}\right]\text{d}z$$

$$+{\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{2{\mathcal{N}}^2/c^2>x^2+4z/c,2a+x<\sqrt{2}\mathcal{N}/c<2b-x\right\}\text{d}z$$

$${\mathcal{P}\mathcal{B}}_2(x,[0,T])=2\Psi (x/\sqrt{2})+\sqrt{2}(T-x)\phi (x/\sqrt{2}),{\mathcal{P}\mathcal{B}}_2(x)=\underset{T\to \infty }{\lim }\frac{{\mathcal{P}\mathcal{B}}_2(x,[0,T])}{T}=\sqrt{2}\phi (x/\sqrt{2}).$$

$$\displaystyle\mathcal{P\!B}_{2}^{h}(x,{E})=\small\left\{\begin{array}[]{ll}e^{(a+x)(a+x-c)}-\nu(a+x,c)+\nu(b-x,c),&a\geq 0,\\ \nu(b-x,c)+\nu(-(a+x),c)+\int_{-\infty}^{0}e^{z}\mathbb{P}\left\{\sqrt{2}|\mathcal{N}|<c,\min(b,t_{2})-\max(a,t_{1})>x\right\}\text{\rm d}z,&a<0,\end{array}\right.$$

$$t_1=\frac{z}{c+\sqrt{2}\mathcal{N}}<0<\frac{-z}{c-\sqrt{2}\mathcal{N}}=t_2,z<0.$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)=\mathbb{E}\left\{\mathbb{I}(\sqrt{2}\mathcal{N}(a+x)+\mathcal{W}>c(a+x{)}^{\lambda })\right\}$$

$$+{\int }_0^{\infty }{e}^z\mathbb{P}\left\{\min (b,t_2)-\max (a,t_1)>x,f_{\mathcal{N}}(t^{\ast })<0,\mathcal{N}>0\right\}\text{d}z.$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)={\int }_{\mathbb{R}}{e}^z\left[\mathbb{P}\left\{f_{\mathcal{N}}(a+x)<0,f_{\mathcal{N}}^{\prime }(a)\ge 0\right\}+\mathbb{P}\left\{f_{\mathcal{N}}(b-x)<0,f_{\mathcal{N}}^{\prime }(b)\le 0\right\}\right]\text{d}z$$

$$+{\int }_{\mathbb{R}}{e}^z\mathbb{P}\left\{\min (b,t_2)-\max (a,t_1)>x,f_{\mathcal{N}}^{\prime }(a)<0<f_{\mathcal{N}}^{\prime }(b),f_{\mathcal{N}}(t^{\ast })<0\right\}\text{d}z.$$

$$C_0(y)=\exp \left((\lambda -1)c{\left|y\right|}^{\lambda }\right),D_0(y,E)={\nu }^{\prime }(b-x,c_0)+{\nu }^{\prime }(-(a+x),-c_0),E=[a,b].$$

$${\mathcal{P}\mathcal{B}}_2^h(x,E)\le \underset{y\in E\setminus \{0\}}{\min }{C}_0(y)D_0(y,E).$$

$$e^{-M}{\mathcal{P}\mathcal{B}}_{\alpha }(x,E)\le {\mathcal{P}\mathcal{B}}_{\alpha }^h(x,E)\le \min ({\mathcal{P}\mathcal{B}}_{\alpha }^h(x,[0,\infty ]),{\mathcal{P}\mathcal{B}}_{\alpha }^h(0,E)),x\ge 0.$$

$${\mathcal{P}\mathcal{B}}_1^h(x)=\left({\int }_0^{\infty }+{\int }_{-\infty }^0\right)e^z\mathbb{P}\left\{Y_z>x\right\}\text{d}z=:I_1(x)+I_2(x).$$

$$\mathbb{P}\left\{Y_z>x\right\}=\mathbb{P}\left\{Y_0>x\right\}\mathbb{P}\left\{\underset{t\ge 0}{\sup }(\sqrt{2}{B}_1(t)-(1+c)t)\ge z\right\},$$

Full text

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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

[TABLE]

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 function $h(t)$ 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 $h(t)=c\left\lvert t\right\rvert^{\lambda}-t^{2}$ 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

AMS Classification: Primary 60G15; secondary 60G70

1. Introduction

Consider a centered Gaussian process $\{X(t),\,t\in\mathbb{R}\}$ with càdlàd sample paths and let for $T>0$

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be the sojourn time of $X$ above the level $u\in\mathbb{R}$ during the observed period $[0,T]$, where $\mathbb{I}\left(\cdot\right)$ stands for the indicator function. In a series of papers culminating in [1], S. Berman derived results on the tail asymptotic behavior of $\nu(u)L_{u,T}$ with an appropriate scaling function $\nu(u)$ such that

[TABLE]

as $u\to\infty$. This essentially builds a bridge of asymptotic behavior of the sojourn time $L_{u,T}$ and the extremal analysis of the Gaussian processes via the link function $C(x)$. However, the asymptotic function $C$ is in general difficult to obtain except very few special processes and approximations have been suggested to evaluate. A related work is given by [2] for a standard Brownian motion with linear drift function. For a stationary and standard Gaussian process $X$ with correlation function $\rho$ satisfies the Pickands’ assumption $\rho(t)=1-|t|^{\alpha}[1+o(1)],\alpha\in(0,2]$ for small $|t|$, [1] showed an explicit form of function $C$ via the following tail distribution (see Theorem 3.3.1 therein)

[TABLE]

where $\mathcal{W}$ is a standard exponential distributed random variable, independent of $B_{\alpha}$, 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

[TABLE]

The recent contribution [3] discussed (1.1) and gave the approximations of the related sojourn time of discrete form for locally stationary Gaussian processes, and [4] investigated general Gaussian processes with strictly positive drift function. For more related discussions on ruin time and the extremal analysis of Gaussian processes and random fields in financial and insurance framework, we refer to [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Motivated by the importance of the crucial function arising in the extremal behavior of the sojourn time and the random processes involved, this paper studies thus a general form of function $G$ given in (1.2). Precisely, define for a compact set $E$ in $\mathbb{R}$ and a continuous drift function $h$ on $E$

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and

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provided that the above integral and limits exist. For $h=0$, we suppress the superscript and write $\mathcal{P\!B}_{\alpha}(x)$ or $\mathcal{P\!B}_{\alpha}(x,E)$. Typical examples of the function $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ can be found in [3] for $h=0,E=\mathbb{R}$, and [4] for polynomial function $h$ and general interval $E$.

Clearly, our setting is very common since $\mathcal{P\!B}_{\alpha}(0)$ is simply the Pickands’ constant, which values are known only for $\alpha=1,2$, i.e., $\mathcal{P\!B}_{1}(0)=1,\mathcal{P\!B}_{2}(0)=1/\sqrt{\pi}$, see e.g., [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] for related studies on its expressions and bounds, while $\mathcal{P\!B}_{\alpha}^{h}(0)$ reduces to the Piterbarg constants for strictly positive drift function. The recent contribution [27] studied the basic properties of the generalized Piterbarg constant ${\mathcal{P\!B}}_{\alpha}^{h}(0)$ for power drift function, which are available for all $\alpha\in(0,2]$. For more general studies on sojourn sets with moving boundary of the processes involved and applications in physics and finance fields, we refer to [28] and among others.

The first result below is concerned with the explicit expression of $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ for the standard Brownian motion and Gaussian distribution, i.e., the fBm with $\alpha=1$ and $\alpha=2$. Here, we focus simply on $E=\mathbb{R}_{+}\cup\{0\}$ and positive drift function $h(t)=ct^{\alpha},t\geq 0,c>0$. In what follows, let $\Psi(\cdot)$ and $\varphi(\cdot)$ be the survival function and probability density function of $\mathcal{N}\sim N(0,1)$, respectively.

Theorem 1.1**.**

Let ${\mathcal{P\!B}}_{\alpha}^{h}(x)$ be the Piterbarg-Berman function given by (1.3) with drift function $h(t)=ct^{\alpha},t\geq 0,c>0$. We have with $x_{c}=(1+c)\sqrt{x/2}$ and $x_{c}^{\prime}=(1-c)\sqrt{x/2}$

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and

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Remark 1.2**.**

The explicit expression of ${\mathcal{P\!B}}_{1}^{h}(x)$ is obtained by the considerable analysis of the stopping time and the random sojourn time involved:

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which nice properties are referred to [29] due to the linear drift function. The general case with non-linear drift function and finite time interval is an open question and it may require definite efforts to develop the initial properties of $\tau_{z}$ and $Y_{z}$ involved.

The main methodology for the establishment of ${\mathcal{P\!B}}_{2}^{h}(x)$ above is essentially determined by the convex curve family $f_{\mathcal{N}}(t)=f_{\mathcal{N}}(t,z),z\in\mathbb{R}$ (recall (1.3))

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since $h(t)=ct^{2}$ and $B_{2}(t)=\mathcal{N}t,\,t\in\mathbb{R}$. In the following theorem, we consider a general drift function $h$ such that $h(t)+t^{2}$ is convex on ${E/\mathbb{R}}$, which leads equivalently that $f_{\mathcal{N}}(t)$ is continuous and convex on $E/\mathbb{R}$. Let thus $s_{1}<s_{2}$ and $t_{1}<t_{2}$ be the two random solutions of $f_{\mathcal{N}}(s)=0,\,s\in E$ and $f_{\mathcal{N}}(t)=0,t\in\mathbb{R}$, respectively if it holds that

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Theorem 1.3**.**

Let $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ be given by (1.3) with $E=[a,b],a<b,a,b\in\mathbb{R}$.

(i) If $h(t)+t^{2}$ is a continuous, convex function on $E$, then

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(ii) If $h(t)+t^{2},t\in\mathbb{R}$ is continuous and convex, and the finite right derivative $h^{\prime}_{+}(a)$ and the left derivative $h^{\prime}_{-}(b)$ exist with finite values, then $h^{\prime}_{+}(a)\leq h^{\prime}_{-}(b)$ and

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Typical examples of Theorem 1.3 are discussed in Propositions 2.1 and 2.2 where we take $h(t)=c|t|^{\lambda}-t^{2}$ with $\lambda=1,2$ and all $\lambda\geq 1$. Some bounds are also specified in Proposition 2.3. All alternative results for $h$ such that $h(t)+t^{2}$ is concave are established, see Theorem 5.1, Propositions 5.1 and 5.2 in Appendix 5.

The study on $\mathcal{P\!B}_{\alpha}^{h}$ for fBms $B_{\alpha}$ with general $\alpha\in(0,1)\cup(1,2)$ is still open since the Slepian’s inequality for extremes of processes is not applicable in the sojourn times setting, see e.g., [13].

The rest of this paper is organized as follows. Section 2 gives several typical examples and its bounds as well. Section 3 carries out a small scale of numerical studies to illustrate the findings. All proofs are relegated to Section 4. We present Appendix 5 for $\mathcal{P\!B}_{2}^{h}$ with concave drift functions.

2. Further Discussions and Applications

Clearly, a straightforward application of Theorem 1.3 with $h(t)=c|t|^{\lambda}-t^{2},\,t\in\mathbb{R},c>0,\lambda\geq 1$ implies the explicit expressions of $\mathcal{P\!B}_{2}^{h}$, which are given in Proposition 2.1 and 2.2 for $\lambda=1,2$ and $\lambda\geq 1$. Some bounds are derived in Proposition 2.3 for $\mathcal{P\!B}_{2}^{h}$.

2.1. Explicit expressions of $\mathcal{P\!B}_{2}^{h}(x,E)$ with $h(t)=c|t|^{\lambda}-t^{2}$

Recalling that $f_{\mathcal{N}}(t)=h(t)+t^{2}-\sqrt{2}\mathcal{N}t+z=f_{-\mathcal{N}}(-t)$ for symmetry $h$ implies that for the sojourn time $\mathcal{L}_{\mathcal{N}}(z)=\int_{a}^{b}\mathbb{I}(f_{\mathcal{N}}(t)<0)\text{\rm d}t$

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We consider only $E=[a,b]$ with $b>0$ unless stated otherwise.

Proposition 2.1**.**

Let $\mathcal{P\!B}_{2}^{h}(x,E)$ be the Piterbarg-Berman function defined in (1.3) with $h(t)=c|t|^{\lambda}-t^{2},c>0$ and $E=[a,b]$. Denote by $f_{\mathcal{N}}(t)=c|t|^{\lambda}-\sqrt{2}\mathcal{N}t+z,\,\mathcal{N}\sim N(0,1)$ as in (1.6).

(i) For $\lambda=2$, we have

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and for $h\equiv 0$,i.e., $c=1$

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(ii) For $\lambda=1$, we have with $\nu(m,c)=e^{m^{2}-c|m|}\Psi\left([c-2m]/{\sqrt{2}}\right)$

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where $t_{1}<t_{2}$ are as in Theorem 1.3 (ii), i.e., the random solutions of $f_{\mathcal{N}}(t)=0,\,t\in\mathbb{R}$ equal

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Intuitively, the three parts involved for $\lambda=2$ above are obtained via the comparison among the symmetric axis $t=\sqrt{2}\mathcal{N}/(2c)$ of the quadratic symmetric curve $f_{\mathcal{N}}(\cdot)$, $a+x/2$ and $b-x/2$. Meanwhile, the well-known Pickands’ constant $\mathcal{P\!B}_{2}(0)=\sqrt{\pi}$ can be implied by $\mathcal{P\!B}_{2}(x)=\sqrt{2}\varphi(x),x\geq 0$. The specification for $\lambda=1$ follows from Theorem 1.3 (ii), and the expression of $\mathcal{P\!B}_{2}^{h}(x,E)$ with $E=[a,b]$ including the original point is more involved due to the piece-wise property of $f_{\mathcal{N}}(\cdot)$. Its detailed expansions are given in Appendix 5.2. Below, we consider the general drift function $h(t)=c|t|^{\lambda}-t^{2}$ with $\lambda\geq 1$. The case with $0<\lambda<1$ is given in Proposition 5.1 in Appendix 5.1.

Proposition 2.2**.**

Let $\mathcal{P\!B}_{2}^{h}(x,E)$ be the Piterbarg-Berman function defined in (1.3) with drift function $h(t)=c|t|^{\lambda}-t^{2},c>0,\lambda\geq 1$ and $E=[a,b]$.

(i) For $a\geq 0$, we have

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(ii) For $a<0$, we have

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Here $t_{1}<t_{2}$ are defined as in Theorem 1.3 (ii), i.e., the two random solutions of $f_{\mathcal{N}}(t)=0,t\in\mathbb{R}$ when its minimum $f_{\mathcal{N}}(t^{*})$ is less than zero.

We see that the Piterbarg-Berman functions require cumbersome calculations even for typical drift functions, see Theorem 1.3, Propositions 2.1 and 2.2. Below, we consider alternatively its bounds.

2.2. Bounds of the Piterbarg-Berman functions

Recalling the geometry property of the convex curve $f_{\mathcal{N}}$, we develop below an upper bound of $\mathcal{P\!B}_{2}^{h}(x,E)$. To simplify the notation, we consider the setting of Proposition 2.2, i.e., $h(t)=c|t|^{\lambda}-t^{2}$ with $c>0,\lambda\geq 1$ (see Proposition 5.2 for the lower bound of $\mathcal{P\!B}_{2}^{h}(x,E)$ with $0<\lambda<1$ in Appendix 5.1). The general bounds $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ for $\alpha\in(0,2]$ are also established by linking the degenerate cases with $h,x,E$ taken into consideration. Set with $c_{0}=c_{0}(y)=\lambda c\left\lvert y\right\rvert^{\lambda-1}\mathrm{sign}(y)$ and $\nu^{\prime}(m,c)=e^{m^{2}-cm}\Psi([c-2m]/\sqrt{2})$

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Proposition 2.3**.**

Let $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ be the Piterbarg-Berman function given by (1.3).

(i) For $\alpha=2$ and $h(t)=c|t|^{\lambda}-t^{2},c>0,\lambda\geq 1$, we have for $E=[a,b]$

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(ii) For $\alpha\in(0,2]$ and drift function $h$ satisfying $M=\max_{s\in E}h(t)<\infty,E\subset\mathbb{R}_{+}\cup\{0\}$. We have

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3. Numerical study

In order to illustrate the theoretical findings in Theorem 1.1, Propositions 2.1 and 2.3, we carry out a small scale of numerical studies for the Piterbarg-Berman function $\mathcal{P\!B}_{\alpha}^{h}(x,E)$.

In Figure 1, we consider $E=[0,\infty]$ for $\mathcal{P\!B}_{1}^{h}(x,E)$. Applying Theorem 1.1 for $h(t)=c|t|$ with different $c$’s, we see that the larger the $c$ is, the more quickly the curve decreases with respect to $x$ (the same as below), and these curves with different $c$’s become closer and closer as $x\to 0$ and $x\to\infty$.

In Figure 2 (and thereafter), we consider finite time interval $E=[a,b]$ and $h(t)=c|t|^{\lambda}-t^{2},\,c>0$ as in Proposition 2.1. Here we take $\lambda=2,E=[0,3],[-2,1],[1,3],[-1,1],[0,4]$ and $c=1,2,4$. We see that $\mathcal{P\!B}_{2}^{h}(x,E),x\in[0,b-a]$ is decreasing slowly.

In Figure 3, we draw $\mathcal{P\!B}_{2}^{h}(x,E),E=[a,b]$ with $h(t)=c|t|-t^{2}$. It seems more sensitive to the time interval $E$ of the same length and decreases strongly as $x\to 0$ but indifferent for larger $x$.

In Figure 4, we apply Proposition 2.3 and show the efficiency of the upper bounds of $\mathcal{P\!B}_{2}^{h}(x,E),E=[a,b]$ with $h(t)=(c-1)t^{2}$ for $c=1,2$. Clearly, the upper bound of the Piterbarg-Berman function become closer to the true values for larger $x$. The error is also determined by the time intervals and the coefficient $c$ of the drift function as well.

Overall, the Piterbarg-Berman function has nice properties with respect to the argument $x$ and the coefficient $c$ arising in the simple drift function $h(t)=c|t|^{\lambda}-t^{2}$. Meanwhile, its essential complexity is closely related to the observed time interval ${E=}[a,b]$ as well as the power index $\lambda$ involved.

4. Proofs

Throughout the proofs, we keep the same notation aforementioned unless stated otherwise. We shall present the proofs of Theorem 1.1 and Propositions 2.1-2.3 one by one.

4.1. Proofs of Theorem 1.1 for $\mathcal{P\!B}_{\alpha}^{h}(x)$ with $h(t)=ct^{\alpha},c>0$ and $\alpha=1,2$

Proof of (i) $\alpha=1$. Recalling the sojourn time $Y_{z}$ given by (1.5), we have

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We deal with the integrals $I_{1}(x)$ and $I_{2}(x)$ according to the upward and downward crossings.

For $I_{1}(x)$. By the lack of upward jumps and the strong Markov property, we have for $z\geq 0$

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where, it follows from [29] that (see e.g., Eq. (3) in p. 255 therein)

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Therefore,

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For $I_{2}(x)$. Note that $\tau_{z}=\inf\{t\geq 0:\sqrt{2}B_{1}(t)-(1+c)t\geq z\}$ for given $z<0$ is a stopping time with cumulative distribution function (cdf) and probability density function (pdf) given by

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by Eq. (3) in page 261 of [29] and the fact that $\varphi(z/\sqrt{2t}+t_{c})=e^{(1+c)z}\varphi(z/\sqrt{2t}-t_{c})$. Therefore, again by the strong Markov property, we have

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where the last step follows since $\tau_{z}$ and $Y_{0}$ are independent. Consequently,

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In the following, we deal with $I_{21}(x)$ and $I_{22}(x)$ subsequently. First, we have by (4.4)

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Similarly, we have by (4.1), (4.4) and (4.1)

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where, with $t_{c}=(1+c)\sqrt{t/2},t_{c}^{\prime}=(1-c)\sqrt{t/2}$,

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Here the three terms $A_{4}^{+}\sim A_{6}^{+}$ for $c>1$ arises since $\Psi(t_{c}^{\prime})=1-\Psi(|t_{c}^{\prime}|)$ as $t_{c}^{\prime}<0$, i.e., $c>1$.

Next, we deal with $I_{22}(x)$ by specifying the three sets of integrals: (i) $A_{1}\sim A_{3}$ with $c>0$; (ii) $A_{4}\sim A_{6}$ with $c\neq 1$ and (iii) $A_{4}^{+}\sim A_{6}^{+}$ with $c>1$.

(ii) For $A_{1}\sim A_{3}$ with $c>0$. By the symmetry of standard normal distributions, we have

[TABLE]

For $A_{3}$, we take $s=\sqrt{t/x},0<t<x$ and thus

[TABLE]

It follows then that

[TABLE]

(ii) For $A_{4}\sim A_{6}$ with $c\neq 1$. We rewrite first $A_{4}$ and $A_{5}$ as follows.

[TABLE]

We decompose further the integrals $A_{4}$ and $B_{4}$ by specifying the integral limits as below

[TABLE]

We deal first with $I_{3},I_{4}$ and $J_{3},J_{4}$. Since $\varphi(|x_{c}|u)e^{cxu^{2}}=\varphi(|x_{c}^{\prime}|u),\varphi(|x_{c}^{\prime}|v)e^{cxv^{2}}=\varphi(|x_{c}|v)$, we have

[TABLE]

and

[TABLE]

For $I_{1},I_{2}$ and $J_{1},J_{2}$, we have

[TABLE]

where $\mu(a)$ and $\nu(a)$ are given by

[TABLE]

Now, for $I_{2}$ and $J_{2}$, we have

[TABLE]

Therefore, summing up $I_{1}\sim I_{4}$ and $J_{1}\sim J_{4}$ to give $A_{4}$ and $B_{4}$ by putting $\mu(\cdot),\nu(\cdot),\omega(\cdot)$ together with the fact that $\nu(a)=-0.5\mathrm{m}(a)-\Psi(a)\omega(a)$ for $\mathrm{m}(\cdot)$ given by (4.1), we have

[TABLE]

Next, we turn to $A_{6}$. Since $\varphi(\sqrt{x_{c}^{2}-t_{c}^{2}})e^{c(x-t)}=\varphi(t_{c}^{\prime})$, we have by the integral of $A_{2}$ given in (4.1)

[TABLE]

Therefore, it follows by $A_{5}=x_{c}^{2}/(cx)A_{4}+B_{4}/(c^{2}x)$ that

[TABLE]

(iii) For $A_{4}^{+}\sim A_{6}^{+}$ with $c>1$. Following similar arguments of $A_{4}\sim A_{6}$, we have

[TABLE]

with

[TABLE]

For $A_{6}^{+}$, by a change of variable $s=t-x$ and $e^{ct}\varphi(t_{c})=\varphi(|t_{c}^{\prime}|)$

[TABLE]

Therefore, we have (recall that $A_{5}^{+}=x_{c}^{2}/(cx)A_{4}^{+}+B_{4}^{+}/(c^{2}x)$)

[TABLE]

Consequently, the desired expression of ${\mathcal{P\!B}}_{\alpha}^{h}(x)$ follows by (4.3), (4.1) and (4.1). Indeed,

[TABLE]

which together with (4.1) implies that

[TABLE]

Now, for $c>1$, we have by (4.1)

[TABLE]

which equals the right-hand side of (4.13) since the summand of the last two terms equals $(1+c)e^{-cx}\mathrm{m}(x_{c}^{\prime})$ using $\mathrm{m}(|a|)+2\omega(|a|)=\mathrm{m}(a),a=x_{c}^{\prime}<0$ (recall (4.1) and (4.1)). Therefore, combining (4.1) for $c=1$, we have a uniform expression of ${\mathcal{P\!B}}_{\alpha}^{h}(x)$ as in (4.13) for all $c>0$. Using again

[TABLE]

and sorting out all terms related to $a=x_{c},x_{c}^{\prime}$, the desired claim for $\alpha=1$ follows.

Proof of (ii) $\alpha=2$. Recall that $f_{\mathcal{N}}(t)=(1+c)t^{2}-\sqrt{2}\mathcal{N}t+z,\,c>0,t\in\mathbb{R}$. Without loss of generality, we consider only the minimum $f_{\mathcal{N}}(t^{*})<0$ with $t^{*}=\mathcal{N}/[\sqrt{2}(1+c)]$. Therefore, there are two solutions $t_{1}<t_{2}$ of the equation $f_{\mathcal{N}}(t)=0$ satisfying

[TABLE]

Consequently,

[TABLE]

where it follows by elementary calculations that $B(x)=\Psi\left((1-c)x/\sqrt{2}\right)e^{-cx^{2}}$ and

[TABLE]

The desired claim for $\alpha=2$ follows. Consequently, we complete the proof of Theorem 1.1. $\Box$

Below, we shall verify Theorem 1.3 by noting that $f_{\mathcal{N}}(t)=[h(t)+t^{2}]-\sqrt{2}\mathcal{N}t+z$ for given $z\in\mathbb{R}$ is continuous and convex if and only if $h(t)+t^{2}$ is. Recall $\mathcal{L}_{\mathcal{N}}(z)=\int_{E}\mathbb{I}\left(f_{\mathcal{N}}(t)<0\right)\text{\rm d}t$.

Proof of Theorem 1.3 (i) We decompose $\{\mathcal{L}_{\mathcal{N}}(z)>x\}$ according to the signs of $f_{\mathcal{N}}(a)$ and $f_{\mathcal{N}}(b)$. Given $z\in\mathbb{R}$, since $f_{\mathcal{N}}({s}),{s}\in E$ is continuous and convex, we have $f_{\mathcal{N}}(a)<0,f_{\mathcal{N}}(b)<0$ implies that $f_{\mathcal{N}}({s})<0,{s}\in[a,b]$. Similar arguments for the cases with $f_{\mathcal{N}}(a)f_{\mathcal{N}}(b)<0$. Finally, for $f_{\mathcal{N}}(a),f_{\mathcal{N}}(b)>0$, suppose without loss of generality that the minimum of $f_{\mathcal{N}}(s),s\in E$ is negative. Therefore, it follows by the convexity of $f_{\mathcal{N}}({s}),{s}\in E$ that, there exist two different roots $s_{1}<s_{2}$ of $f_{\mathcal{N}}(s)=0,s\in E$. Therefore, $\{s\in E:f_{\mathcal{N}}(s)<0\}=(s_{1},s_{2})$. Consequently, the first desired claim follows.

(ii) It follows by the convexity of $f_{\mathcal{N}}(t)$ that $f_{\mathcal{N}}^{\prime}(a):=h^{\prime}_{+}(a)+2a-\sqrt{2}\mathcal{N}\leq f_{\mathcal{N}}^{\prime}(b):=h^{\prime}_{-}(b)+2b-\sqrt{2}\mathcal{N}$. Therefore, we decompose $\{\mathcal{L}_{\mathcal{N}}(z)>x\}$ according to the three cases that $0\leq f_{\mathcal{N}}^{\prime}(a)<f_{\mathcal{N}}^{\prime}(b)$, $f_{\mathcal{N}}^{\prime}(a)<0<f_{\mathcal{N}}^{\prime}(b)$ and $f_{\mathcal{N}}^{\prime}(a)<f_{\mathcal{N}}^{\prime}(b)\leq 0$. For the first case with $f_{\mathcal{N}}^{\prime}(a)\geq 0$, it follows by the convexity of $f_{\mathcal{N}}(t),t\in\mathbb{R}$ that

[TABLE]

implying that $\{\mathcal{L}_{\mathcal{N}}(z)>x,f_{\mathcal{N}}^{\prime}(a)>0\}=\{f_{\mathcal{N}}(a+x)<0,f_{\mathcal{N}}^{\prime}(a)>0\}$. Similar arguments apply for the other two cases. We complete the proof of Theorem 1.3. $\Box$

4.2. Proofs of Propositions 2.1 and 2.2

In Proposition 2.1, we take $h(t)=c|t|^{\lambda}-t^{2}$ with $\lambda=1,2$. The general case with $\lambda\geq 1$ is shown in Proposition 2.2.

(i) For $\lambda=2$. Suppose without loss of generality that the minimum of $f_{\mathcal{N}}(t),t\in\mathbb{R}$ is negative and the two solutions $t_{1}<t_{2}$ of $f_{\mathcal{N}}(t)=0,{t}\in\mathbb{R}$ satisfying

[TABLE]

We have thus (recall (1.3))

[TABLE]

implying that (note that $b-a\geq x\geq 0$)

[TABLE]

with

[TABLE]

Therefore, the remaining argument follows by elementary calculations and the claim is obtained.

(ii) For $\lambda=1$. We have $f_{\mathcal{N}}({t})=c|{t}|-\sqrt{2}\mathcal{N}{t}+z$, a continuous and piece-wise linear function such that

[TABLE]

Therefore, we consider below the two cases with $0\leq a<b$ and $a<0<b$.

As $0\leq a<b$. Clearly, the function $f_{\mathcal{N}}({t}),{t}\geq 0$ is linear with slope $f_{\mathcal{N}}^{\prime}(a)=f_{\mathcal{N}}^{\prime}(b)=c-\sqrt{2}\mathcal{N}$. Recalling $b-a\geq x$ and $f_{\mathcal{N}}(0)=z$, we have with $\nu(\cdot,\cdot)$ given in the theorem

[TABLE]

following elementary calculations.

As $a<0<b$. It follows by Theorem 1.3 (ii) that, with slope $-c-\sqrt{2}\mathcal{N}=f_{\mathcal{N}}^{\prime}(a)<f_{\mathcal{N}}^{\prime}(b)=c-\sqrt{2}\mathcal{N}$

[TABLE]

where $t_{1}<0<t_{2}$ are given by Proposition 2.1. For the purpose of the explicit expressions of the events involved, we rewrite the first integral based on the sign of $a+x$

[TABLE]

Similarly, the second integral satisfies $H_{2}=\int_{\mathbb{R}}e^{z}\mathbb{P}\left\{f_{\mathcal{N}}(b-x)<0,f_{\mathcal{N}}^{\prime}(b)<0\right\}\text{\rm d}z=\nu(b-x,c).$ Consequently, the desired claims of Proposition 2.1 are obtained. $\Box$

In view of (1.3), we have $f_{\mathcal{N}}(s)=c|t|^{\lambda}-\sqrt{2}\mathcal{N}t+z$ is convex such that

[TABLE]

Proof of Proposition 2.2 Clearly, it follows from Theorem 1.3 (ii) that, the claim for $a<0<b$ holds since $h^{\prime}_{+}(a)+2a=-\lambda c(-a)^{\lambda-1}<0<h_{-}^{\prime}(b)+2b=\lambda cb^{\lambda-1}$ exist with finite values. For $0\leq a<b$, it follows from (5.1) that

[TABLE]

Consequently,

[TABLE]

where the first integral is $\mathbb{E}\left\{\mathbb{I}(\sqrt{2}\mathcal{N}(a+x)+\mathcal{W}>c(a+x)^{\lambda})\right\}$ by a change of variable $z^{\prime}=-z$. We compete the proof of Proposition 2.2. $\Box$

4.3. Proof of Proposition 2.3 for the bounds of $\mathcal{P\!B}_{\alpha}^{h}$

We start with the drift function $h(t)=c|t|^{\lambda}-t^{2}$ and $\alpha=2$, and then for the general bounds available for $\alpha\in(0,2]$.

(i) Bounds of $\mathcal{P\!B}_{2}^{h}$ for $h(t)=c|t|^{\lambda}-t^{2}$ with $\lambda\geq 1$. Clearly, $g(t)=c\left\lvert t\right\rvert^{\lambda},t\in[a,b]$ and $\lambda\geq 1$ is convex and thus

[TABLE]

Therefore, with $c_{0}:=c_{0}(y)=g^{\prime}(y){=\lambda c\left\lvert y\right\rvert^{\lambda-1}\mathrm{sign}(y)},y\in{E}\setminus\{0\}$

[TABLE]

Next, we verify $D_{0}(y)$ satisfies (2.1) by considering simply $\tilde{f}_{\mathcal{N}}(t)=(c_{0}-\sqrt{2}\mathcal{N})t+z$. Indeed, we have $\tilde{f}_{\mathcal{N}}^{\prime}(a)=c_{0}-\sqrt{2}\mathcal{N},c_{0}\in\mathbb{R}$ and $\tilde{f}_{\mathcal{N}}(0)=z$. Therefore,

[TABLE]

where the first integral equals

[TABLE]

Similarly, the second integral equals $\nu^{\prime}(b-x,c_{0})$. The claim follows by the arbitrary of $y\in[a,b]\setminus\{0\}$.

(ii) Bounds for $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ with $\alpha\in(0,2]$. Recalling $M=\max_{s\in E}h(t)$, we have

[TABLE]

Hence,

[TABLE]

Conversely, the sojourn time $\mathcal{L}_{\alpha}^{h}(z,E)$ is increasing with respect to $T$ involved in the time interval $E=[0,T]$. We have

[TABLE]

implying that $\mathcal{P\!B}_{\alpha}^{h}(x,E)\leq\mathcal{P\!B}_{\alpha}^{h}(x,[0,\infty])$. Meanwhile, it is clear that $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ is decreasing with respect to $x\geq 0$. The claim follows. We complete the proof of Proposition 2.3. $\Box$

Acknowledgements  C. Ling would like to thank Prof. Krzysztof Dȩbicki for several useful discussions and important comments during the work on the contribution. C. Ling is supported by the National Natural Science Foundation (NSNF) (11604375). H. Zhang is partially supported by the NSNF (11701469) and the Basic and Frontier Research Program of Chongqing, China (cstc2016jcyjA0510).

5. Appendix

In Section 5.1, we discuss first $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ with drift function such that $h(t)+t^{2}$ is continuous and concave in Theorem 5.1, which is illustrated by Proposition 5.1 with $h(t)=c\left\lvert t\right\rvert^{\lambda}-t^{2},0<\lambda<1$ as well as its lower bounds in Proposition 5.2. Second, we present in Section 5.2 for the detailed calculations of Proposition 2.1.

5.1. Discussions on $\mathcal{P\!B}_{\alpha}^{h}$ with $h(t)+t^{2}$ being concave

Recall the curve family $f_{\mathcal{N}}(t)=f_{\mathcal{N}}(t,z),z\in\mathbb{R}$ (recall (1.3)) and the sojourn time $\mathcal{L}_{\mathcal{N}}(z)$ given by

[TABLE]

Thus $f_{\mathcal{N}}(t)$ is continuous and concave if and only if $h(t)+t^{2}$ is. Let thus $s_{1}<s_{2}$ and $t_{1}<t_{2}$ be the two random solutions of $f_{\mathcal{N}}(s)=0,\,s\in E$ and $f_{\mathcal{N}}(t)=0,t\in\mathbb{R}$, respectively if it holds that

[TABLE]

Theorem 5.1**.**

Let $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ be given by (1.3) with $E=[a,b],a<b,a,b\in\mathbb{R}$.

(i) If $h(t)+t^{2}$ is a continuous, concave function on $E$, then

[TABLE]

(ii) If $h(t)+t^{2},t\in\mathbb{R}$ is continuous and concave, and the finite right derivative $h^{\prime}_{+}(a)$ and the left derivative $h^{\prime}_{-}(b)$ exist with finite values, then $h^{\prime}_{+}(b)\leq h^{\prime}_{-}(a)$ and

[TABLE]

Proof. (i) We decompose $\{\mathcal{L}_{\mathcal{N}}(z)>x\}$ according to the signs of $f_{\mathcal{N}}(a)$ and $f_{\mathcal{N}}(b)$. Given $z\in\mathbb{R}$, since $f_{\mathcal{N}}({s}),{s}\in E$ is continuous and concave, we have the maximum $f_{\mathcal{N}}(s^{*})\leq 0,s^{*}\in E$ implies that $f_{\mathcal{N}}({s})\leq 0,{s}\in E$. While $f_{\mathcal{N}}(s^{*})>0$, we analyze the three cases with $f_{\mathcal{N}}(a)<0,f_{\mathcal{N}}(b)\geq 0$; $f_{\mathcal{N}}(a)\geq 0,f_{\mathcal{N}}(b)<0$; and $f_{\mathcal{N}}(a),f_{\mathcal{N}}(b)<0$. For the first case, we have by the concavity

[TABLE]

and if there is a $t_{0}\in[a+x,b)$ such that $f_{\mathcal{N}}(t_{0})=0$, then it holds that $f_{\mathcal{N}}(t)\geq 0$ for all $t\in[a+x,b)$. Therefore, $\{\mathcal{L}_{\mathcal{N}}(z)>x,f_{\mathcal{N}}(a)<0,f_{\mathcal{N}}(b)\geq 0\}=\{f_{\mathcal{N}}(a+x)<0,f_{\mathcal{N}}(b)\geq 0\}$. Similar argument applies for the second case. Finally, for $f_{\mathcal{N}}(a),f_{\mathcal{N}}(b)<0$ and $f_{\mathcal{N}}(s^{*})>0,s\in E$, it follows by the concavity of $f_{\mathcal{N}}({s}),{s}\in E$ that, there exist two different roots $s_{1}<s_{2}$ of $f_{\mathcal{N}}(s)=0,s\in E$. Therefore, $\{s\in E:f_{\mathcal{N}}(s)<0\}=(a,s_{1})\cup(s_{2},b)$. Consequently, the first desired claim follows.

(ii) For the first case, $f_{\mathcal{N}}(t^{*})$ is non-negative implies that $f_{\mathcal{N}}({t})\leq 0,t\in\mathbb{R},{z\in(-\infty,-h(0))}$. The rest cases follow by the concavity of $f_{\mathcal{N}}(t)$ that $f_{\mathcal{N}}^{\prime}(a):=h^{\prime}_{+}(a)+2a-\sqrt{2}\mathcal{N}\geq f_{\mathcal{N}}^{\prime}(b):=h^{\prime}_{-}(b)+2b-\sqrt{2}\mathcal{N}$. Therefore, we decompose $\{\mathcal{L}_{\mathcal{N}}(z)>x\}$ according to the three cases that $f_{\mathcal{N}}^{\prime}(b)\geq 0$, $f_{\mathcal{N}}^{\prime}(b)<0<f_{\mathcal{N}}^{\prime}(a)$ and $f_{\mathcal{N}}^{\prime}(a)\leq 0$. For the case with $f_{\mathcal{N}}^{\prime}(b)\geq 0$, it follows by the concavity of $f_{\mathcal{N}}(t),t\in\mathbb{R}$ that

[TABLE]

implying that $\{\mathcal{L}_{\mathcal{N}}(z)>x,f_{\mathcal{N}}^{\prime}(b)\geq 0\}=\{f_{\mathcal{N}}(a+x)<0,f_{\mathcal{N}}^{\prime}(b)\geq 0\}$. Similar arguments apply for the other two cases. We complete the proof of Theorem 5.1. $\Box$

Proposition 5.1**.**

Let $\mathcal{P\!B}_{2}^{h}(x,E)$ be the Piterbarg-Berman function defined in (1.3) with drift function $h(t)=c|t|^{\lambda}-t^{2},c>0,0<\lambda<1$ and $E=[a,b]$.

(i) For $a\geq 0$, we have

[TABLE]

(ii) For $a<0$, we have

[TABLE]

Here $t_{1}<t_{2}$ are the solutions of $f_{\mathcal{N}}(t)=0,t\in\mathbb{R}$ as its maximum $f_{\mathcal{N}}(t^{*})$ is greater than zero.

Proof. Clearly, it follows from Theorem 5.1 (ii) that, the claim for $a<0<b$ holds since $h^{\prime}_{+}(a)+2a=-\lambda c(-a)^{\lambda-1}<0<h_{-}^{\prime}(b)+2b=\lambda cb^{\lambda-1}$ exist with finite values. For $0\leq a<b$, since

[TABLE]

we have

[TABLE]

We compete the proof of Proposition 5.1. $\Box$

Proposition 5.2**.**

Let $\mathcal{P\!B}_{\alpha}^{h}(x,E)$ be the Piterbarg-Berman function given by (1.3).

For $\alpha=2$ and $h(t)=c|t|^{\lambda}-t^{2},c>0,0<\lambda<1$, we have for $E=[a,b]$

[TABLE]

Here $C_{0}(y)$ and $D_{0}(y)$ are as in (2.1).

Proof. The main arguments are similar to those for Proposition 2.3. Let $g(t)=c\left\lvert t\right\rvert^{\lambda},t\in[a,b],0<\lambda<1$, which is continuous and concave. Therefore, instead of (4.15), we have

[TABLE]

with $y\in{E}\setminus\{0\}$ and $yt\geq 0$.

First, for $a\geq 0$, we have by a change of variable $z^{\prime}=z-c_{0}y+g(y)$

[TABLE]

Next, we deal with the case of $a<0<b$. Clearly, we have

[TABLE]

where the second inequality holds by (5.1). Similar argument yields that

[TABLE]

Consequently,

[TABLE]

By the arbitrary of $y$, we complete the proof of Proposition 5.2. $\Box$

5.2. Explicit expressions of integrals involved in $\mathcal{P\!B}_{2}^{h}(x,{E})$ in Proposition 2.1

(i) For $\lambda=2$. Denote by $f_{\mathcal{N}}(t)=ct^{2}-\sqrt{2}\mathcal{N}t+z$ and

[TABLE]

where the relationship between $I_{1},I_{2}$ follows by the symmetry of $\mathcal{N}$. According to the sign of $a+x$,

[TABLE]

Finally, we deal with $I_{3}$ below. Rewriting $\{2\mathcal{N}^{2}/c^{2}>x^{2}+4z/c\}$ according to $x^{2}+4z/c\geq,<0$, we have by elementary calculations

[TABLE]

One can further write down the related integrals with respect to $z$ by specifying the maxima/minima involved with restriction of the left-endpoint being less than the corresponding right-endpoint. Consequently, sum up all related terms given in (5.2)$\sim$(5.2) and thus the explicit expression of $\mathcal{P\!B}_{2}^{h}(x,{E})$ is obtained.

Explicit expression of $\mathcal{P\!B}_{2}(x,[0,T])$. Replace $a=0,b=T$ and $c=1,x\in[0,T]$ in the expressions of $I_{1},I_{2}$ and $I_{3}$ above. First,

[TABLE]

where the integral equals

[TABLE]

Consequently, we have $I_{1}(0,x)=\Psi(x/\sqrt{2}),\,x\geq 0$.

Now, for $I_{2}(T,x)$, we have (recall $-T+x<0$)

[TABLE]

where the first integral is (set $m:=T-x$)

[TABLE]

Thus, $I_{2}(T,x)=\Psi(x/\sqrt{2}),\,x\geq 0$.

Now, we analyze $I_{3}$ as below

[TABLE]

which together with $I_{1},I_{2}$ implies the desired result of $\mathcal{P\!B}_{2}(x,[0,T])$.

(ii) For $\lambda=1$. We specify the following integral

[TABLE]

First, we rewrite $H_{3}$ by comparing $t_{i},\,i=1,2$ and $a,b$, we have

[TABLE]

where

[TABLE]

We deal with the four parts $H_{31}\sim H_{34}$ one by one. Indeed,

[TABLE]

where $\beta$ is given by

[TABLE]

Similar argument for $H_{32}$ implies that (set below $\alpha=z/a+z/b-\beta$)

[TABLE]

Next, for $H_{33}$, we have

[TABLE]

and similar argument for $H_{34}$ implies that

[TABLE]

Consequently, we give the explicit calculations of $H_{3}$ by the related claims of $H_{31}\sim H_{34}$.

$\Box$