# Sojourn Ruin of a Two-Dimensional Fractional Bronwian Motion Risk Process

Grigori Jasnovidov  Grigori Jasnovidov, Department of Actuarial Science,
University of Lausanne,
UNIL-Dorigny, 1015 Lausanne, Switzerland<EMAIL_ADDRESS>
Abstract: This paper derives the asymptotic behavior of

$\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}B_{H}(s)-c_{1}s>q_{1}u,B_{H}(s)-c_{2}s>q_{2}u\Big{)}ds>T_{u}\right\\},\quad
u\to\infty,$

where $B_{H}$ is a fractional Brownian motion, $c_{1},c_{2},q_{1},q_{2}>0,\
H\in(0,1),\ T_{u}\geq 0$ is a measurable function and $\mathbb{I}(\cdot)$ is
the indicator function.

Key Words: fractional Brownian motion; simultaneous ruin probability; two-
dimensional risk processes; sojourn ruin;

AMS Classification: Primary 60G15; secondary 60G70

##  1\. Introduction & Preliminaries

Consider the risk model defined by

(1) $\displaystyle R(t)=u+\rho t-X(t),\ \ \ \ t\geq 0,$

where $X(t)$ is a centered Gaussian risk process with a.s. continuous sample
paths, $\rho>0$ is the net profit rate and $u>0$ is the initial capital. This
model is relevant to insurance and financial applications, see, e.g., [13]. A
question of numerous investigations (see [12, 14, 19, 15, 8, 10, 7, 5, 3, 6,
1, 4, 24, 23, 25, 22]) is the study of the asymptotics of the classical ruin
probability

(2) $\displaystyle\lambda(u):=\mathbb{P}\left\\{\exists t\geq
0:R(t)<0\right\\}$

as $u\to\infty$ under different levels of generality. It turns out, that only
for $X$ being a Brownian motion (later on BM) $\lambda(u)$ can be calculated
explicitly: if $X$ is a standard BM, then $\lambda(u)=e^{-2\rho u},\
u,\rho>0,$ see [11]. Since it seems impossible to find the exact value of
$\lambda(u)$ in other cases, the approximations of $\lambda(u)$ as
$u\to\infty$ is dealt with. Some contributions (see, e.g., [17, 9]), extend
the classical ruin problem to the so-called sojourn ruin problem, i.e.,
approximation of the sojourn ruin probability defined by

(3)
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(R(s)<0)ds>T_{u}\right\\},$

where $T_{u}\geq 0$ is a measurable function of $u$. As in the classical case,
only for $X$ being a BM the probability above can be calculated explicitly,
see [9]:

$\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(B(s)-cs>u)ds>T\right\\}=\Big{(}2(1+c^{2}T)\Psi(c\sqrt{T})-\frac{c\sqrt{2T}}{\sqrt{\pi}}e^{\frac{-c^{2}T}{2}}\Big{)}e^{-2cu},\
\ c>0,\ T,u\geq 0,$

where $\Psi$ is the survival function of a standard Gaussian random variable,
$B$ is a standard BM and $\mathbb{I}(\cdot)$ is the indicator function.
Motivated by [18] (see also [15, 16]), we study a generalization of the main
problem in [18] for the sojourn ruin, i.e., we shall study the asymptotics of

$\mathbb{C}_{T_{u}}(u):=\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}B_{H}(s)-c_{1}s>q_{1}u,B_{H}(s)-c_{2}s>q_{2}u\Big{)}ds>T_{u}\right\\},\
\ \ \ u\to\infty,$

where $B_{H}$ is a standard fractional Brownian motion (later on fBM), i.e., a
Gaussian process with zero expectation and covariance defined by

$\displaystyle\operatorname{cov}(B_{H}(s),B_{H}(t))=\frac{|t|^{2H}+|s|^{2H}-|t-s|^{2H}}{2},\
\ \ \ \ \ t,s\in\mathbb{R}.$

The ruin probability above is of interest for reinsurance models, see [18] and
references therein. By the self-similarity of fBM we have

$\displaystyle\mathbb{C}_{T_{u}}(u)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}B_{H}(su)>c_{1}su+q_{1}u,B_{H}(su)>c_{2}su+q_{2}u\Big{)}d(su)>T_{u}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}u^{H}B_{H}(s)>(c_{1}s+q_{1})u,u^{H}B_{H}(s)>(c_{2}s+q_{2})u\Big{)}ds>T_{u}/u\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\left(\frac{B_{H}(s)}{\max(c_{1}s+q_{1},c_{2}s+q_{2})}>u^{1-H}\right)ds>T_{u}/u\right\\}.$

In order to prevent the problem of degenerating to the one-dimensional sojourn
problem discussed in [17, 9] (i.e., to impose the denominator in the line
above be nonlinear function) we assume that

(4) $\displaystyle c_{1}>c_{2},\ \ \ q_{2}>q_{1}.$

The variance of the process two lines above can achieve its unique maxima only
at one of the following points:

(5) $\displaystyle t_{*}=\frac{q_{2}-q_{1}}{c_{1}-c_{2}},\ \ \
t_{1}=\frac{q_{1}H}{(1-H)c_{1}},\ \ \ t_{2}=\frac{q_{2}H}{(1-H)c_{2}}.$

From (4) it follows that $t_{1}<t_{2}$; as we shall see later, the order
between $t_{1},t_{2}$ and $t_{*}$ determines the asymptotics of
$\mathbb{C}_{T_{u}}(u)$ as $u\to\infty$. As mentioned in [5], for the
approximation of the one-dimensional Parisian ruin probability we need to
control the growth of $T_{u}$ as $u\to\infty$. As in [5], we impose the
following condition:

(6) $\displaystyle\lim_{u\to\infty}T_{u}u^{1/H-2}=T\in[0,\infty),\ H\in(0,1).$

Note that $T_{u}$ satisfying (6) may go to $\infty$ for $H>1/2$, converges to
non-negative limit for $H=1/2$ and approaches $0$ for $H<1/2$ as $u\to\infty$.
We see later on in Proposition 2.2 that the condition above is necessary and
it seems very difficult to derive the exact asymptotics of
$\mathbb{C}_{T_{u}}(u)$ as $u\to\infty$ without it.

The rest of the paper is organized in the following way. In the next section
we present the main results of the paper, in Section 3 we give all proofs,
while technical calculations are deferred to the Appendix.

##  2\. Main Results

Define for some function $h$ and $K\geq 0$ the sojourn Piterbarg constant by

$\displaystyle\mathcal{B}^{h}_{K}=\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{-\infty}^{\infty}\mathbb{I}\Big{(}\sqrt{2}B(s)-|s|+h(s)>x\Big{)}ds>K\right\\}e^{x}dx$

when the integral above is finite and Berman’s constant by

$\mathcal{B}_{2H}(x)=\lim_{S\to\infty}\frac{1}{S}\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{0}^{S}\mathbb{I}(\sqrt{2}B_{H}(t)-t^{2H}+z>0)dt>x\right\\}e^{-z}dz,\quad
x\geq 0.$

It is known (see, e.g., [9]) that $\mathcal{B}_{2H}(x)\in(0,\infty)$ for all
$x\geq 0$; we refer to [9] and references therein for the properties of
relevant Berman’s constants. Let for $i=1,2$

(7) $\displaystyle\ \ \mathbb{D}_{H}=\frac{c_{1}t_{*}+q_{1}}{t_{*}^{H}},\
K_{H}=\frac{2^{\frac{1}{2}-\frac{1}{2H}}\sqrt{\pi}}{\sqrt{H(1-H)}},\
\mathbb{C}_{H}^{(i)}=\frac{c_{i}^{H}q_{i}^{1-H}}{H^{H}(1-H)^{1-H}},\
D_{i}=\frac{c_{i}^{2}(1-H)^{2-\frac{1}{H}}}{2^{\frac{1}{2H}}H^{2}}.$

Now we are ready to give the asymptotics of $\mathbb{C}_{T_{u}}(u)$ as
$u\to\infty$.

######  Theorem 2.1.

Assume that (4) holds and $T_{u}$ satisfies (6).
1)If $t_{*}\notin(t_{1},t_{2})$, then as $u\to\infty$

(8)
$\displaystyle\mathbb{C}_{T_{u}}(u)\sim(\frac{1}{2})^{\mathbb{I}(t_{*}=t_{i})}\times\begin{cases}\Big{(}2(1+c_{i}^{2}T)\Psi(c_{i}\sqrt{T})-\frac{c_{i}\sqrt{2T}}{\sqrt{\pi}}e^{\frac{-c_{i}^{2}T}{2}}\Big{)}e^{-2c_{i}q_{i}u},&H=1/2\\\
K_{H}\mathcal{B}_{2H}(TD_{i})(\mathbb{C}_{H}^{(i)}u^{1-H})^{\frac{1}{H}-1}\Psi(\mathbb{C}_{H}^{(i)}u^{1-H}),&H\neq
1/2,\end{cases}$

where $i=1$ if $t_{*}\leq t_{1}$ and $i=2$ if $t_{*}\geq t_{2}$.

2) If $t_{*}\in(t_{1},t_{2})$ and $\lim\limits_{u\to\infty}T_{u}u^{2-1/H}=0$
for $H>1/2$, then as $u\to\infty$

(9)
$\displaystyle\mathbb{C}_{T_{u}}(u)\sim\Psi(\mathbb{D}_{H}u^{1-H})\times\begin{cases}1,&H>1/2\\\
\mathcal{B}_{T^{\prime}}^{d},&H=1/2\\\
\mathcal{B}_{2H}(\overline{D}T)Au^{(1-H)(1/H-2)},&H<1/2,\end{cases}$

where $\mathcal{B}_{T^{\prime}}^{d}\in(0,\infty)$,

(10) $\displaystyle
T^{\prime}=T\frac{(c_{1}q_{2}-q_{1}c_{2})^{2}}{2(c_{1}-c_{2})^{2}},\ \
d(s)=s\frac{c_{1}q_{2}+c_{2}q_{1}-2c_{2}q_{2}}{c_{1}q_{2}-q_{1}c_{2}}\mathbb{I}(s<0)+s\frac{2c_{1}q_{1}-c_{1}q_{2}-q_{1}c_{2}}{c_{1}q_{2}-q_{1}c_{2}}\mathbb{I}(s\geq
0)$

and

(11)
$\displaystyle\\!\\!A=\Big{(}|H(c_{1}t_{*}+q_{1})-c_{1}t_{*}|^{-1}+|H(c_{2}t_{*}+q_{2})-c_{2}t_{*}|^{-1}\Big{)}\frac{t_{*}^{H}\mathbb{D}_{H}^{\frac{1}{H}-1}}{2^{\frac{1}{2H}}},\
\ \
\overline{D}=\frac{(c_{1}t_{*}+q_{1})^{\frac{1}{H}}}{2^{\frac{1}{2H}}t_{*}^{2}}.$

Note that if $T=0$, then the result above reduces to Theorem 3.1 in [18]. As
already mentioned in the introduction (6) is a necessary condition for the
theorem above. To illustrate situation when it is not satisfied we consider a
”simple” scenario with $T_{u}$ being a positive constant.

######  Proposition 2.2.

If $H<1/2,\ T_{u}=T>0$ and $t_{*}\in(t_{1},t_{2})$, then

$\displaystyle\bar{C}\Psi(\mathbb{D}_{H}u^{1-H})e^{-C_{1,\alpha}u^{2-4H}-C_{2,\alpha}u^{2(1-3H)}}$
$\displaystyle\leq$ $\displaystyle\mathbb{C}_{T_{u}}(u)$ $\displaystyle\leq$
$\displaystyle(2+o(1))\Psi(\mathbb{D}_{H}u^{1-H})\Psi\left(u^{1-2H}\frac{T^{H}\mathbb{D}_{H}}{2t_{*}^{H}}\right),\quad
u\to\infty,$

where $\bar{C}\in(0,1)$ is a fixed constant that does not depend on $u$ and

$\displaystyle\alpha=\frac{T^{2H}}{2t_{*}^{2H}},\ \ \ \ \ \
C_{i,\alpha}=\frac{\alpha^{i}}{i}\mathbb{D}_{H}^{2},\ \ \ i=1,2.$

Note that the lower bound in the proposition above decays to zero
exponentially faster than the upper bound as $u\to\infty$.

##  3\. Proofs

First we give the following auxiliary results. As shown, e.g., in Lemma 2.1 in
[21]

(12)
$\displaystyle(1-\frac{1}{u^{2}})\frac{1}{\sqrt{2\pi}u}e^{-u^{2}/2}\leq\Psi(u)\leq\frac{1}{\sqrt{2\pi}u}e^{-u^{2}/2},\
\ \ u>0.$

Recall that $K_{H},D_{1}$ and $\mathbb{C}_{H}^{(1)}$ are defined in (7). A
proof of the proposition below is given in the Appendix.

######  Proposition 3.1.

Assume that $T_{u}$ satisfies (6). Then as $u\to\infty$

$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(B_{H}(t)-c_{1}t>q_{1}u)dt>T_{u}\right\\}\sim\begin{cases}\Big{(}2(1+c_{1}^{2}T)\Psi(c_{1}\sqrt{T})-\frac{c_{1}\sqrt{2T}}{\sqrt{\pi}}e^{\frac{-c_{1}^{2}T}{2}}\Big{)}e^{-2c_{1}q_{1}u},&H=1/2,\\\
K_{H}\mathcal{B}_{2H}(TD_{1})(\mathbb{C}_{H}^{(1)}u^{1-H})^{\frac{1}{H}-1}\Psi(\mathbb{C}_{H}^{(1)}u^{1-H}),&H\neq
1/2.\end{cases}$

Now we are ready to perform our proofs.

Proof of Theorem 2.1. Case (1). Assume that $t_{*}<t_{1}$. Let

$\displaystyle V_{i}(t)=\frac{B_{H}(t)}{c_{i}t+q_{i}}\ \ \ \text{ and }\ \ \
\psi_{i}(T_{u},u)=\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(B_{H}(t)-c_{i}t>q_{i}u)ds>T_{u}\right\\},\
\ \ i=1,2.$

For $0<\varepsilon<t_{1}-t_{*}$ by the self-similarity of fBM we have

$\displaystyle\psi_{1}(T_{u},u)\geq\mathbb{C}_{T_{u}}(u)\geq\mathbb{P}\left\\{\int\limits_{t_{1}-\varepsilon}^{t_{1}+\varepsilon}\mathbb{I}(V_{1}(t)>u^{1-H},V_{2}(t)>u^{1-H})dt>\frac{T_{u}}{u}\right\\}=\mathbb{P}\left\\{\int\limits_{t_{1}-\varepsilon}^{t_{1}+\varepsilon}\mathbb{I}(V_{1}(t)>u^{1-H})dt>\frac{T_{u}}{u}\right\\}.$

We have by Borel-TIS inequality, see [24] (details are in the Appendix)

(13)
$\displaystyle\psi_{1}(T_{u},u)\sim\mathbb{P}\left\\{\int\limits_{t_{1}-\varepsilon}^{t_{1}+\varepsilon}\mathbb{I}(V_{1}(t)>u^{1-H})ds>T_{u}/u\right\\},\
\ \ u\to\infty$

implying $\mathbb{C}_{T_{u}}(u)\sim\psi_{1}(T_{u},u)$ as $u\to\infty$. The
asymptotics of $\psi_{1}(T_{u},u)$ is given in Proposition 3.1, thus the claim
follows.

Assume that $t_{*}=t_{1}$. We have

$\displaystyle\mathbb{P}\left\\{\int\limits_{t_{1}}^{\infty}\mathbb{I}(V_{1}(s)>u^{1-H})ds>T_{u}\right\\}$
$\displaystyle\leq$ $\displaystyle\mathbb{C}_{T_{u}}(u)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left\\{\int\limits_{t_{1}}^{\infty}\mathbb{I}(V_{1}(s)>u^{1-H})ds>T_{u}\right\\}+\mathbb{P}\left\\{\\!\exists
t\\!\in\\![0,t_{1}]:V_{2}(t)\\!>\\!u^{1-H}\right\\}.$

From the proof of Theorem 3.1, case (4) in [18] it follows that the second
term in the last line above is negligible comparing with the final asymptotics
of $\mathbb{C}_{T_{u}}(u)$ given in (8), hence

$\mathbb{C}_{T_{u}}(u)\sim\mathbb{P}\left\\{\int\limits_{t_{1}}^{\infty}\mathbb{I}(V_{1}(s)>u^{1-H})ds>T_{u}\right\\},\quad
u\to\infty.$

Since $t_{1}$ is the unique maxima of $\text{Var}\\{V_{1}(t)\\}$ from the
proof of Theorem 2.1, case i) in [9] we have

$\displaystyle\mathbb{P}\left\\{\int\limits_{t_{1}}^{\infty}\mathbb{I}(V_{1}(t)>u^{1-H})dt>T_{u}/u\right\\}$
$\displaystyle\sim$
$\displaystyle\frac{1}{2}\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(V_{1}(t)>u^{1-H})dt>T_{u}/u\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(B_{H}(t)-c_{1}t>q_{1}u)dt>T_{u}\right\\},\
\ u\to\infty.$

The asymptotics of the last probability above is given in Proposition 3.1
establishing the claim. Case $t_{*}\geq t_{2}$ follows by the same arguments.

Case (2). Assume that $H>1/2$. We have by Theorem 2.1 in [16] and Theorem 3.1
in [18] with

$\displaystyle\mathcal{R}_{T_{u}}(u)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\exists t\geq
0:B_{H}(t)-c_{1}t>q_{1}u,B_{H}(t)-c_{2}t>q_{2}u\right\\},$
$\displaystyle\mathcal{P}_{T_{u}}(u)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\exists t\geq
0:\inf\limits_{s\in[t,t+T_{u}]}(B_{H}(s)-c_{1}s)>q_{1}u,\inf\limits_{s\in[t,t+T_{u}]}(B_{H}(s)-c_{2}s)>q_{2}u\right\\}$

that

$\Psi(\mathbb{D}_{H}u^{1-H})\sim\mathcal{P}_{T_{u}}(u)\leq\mathbb{C}_{T_{u}}(u)\leq\mathcal{R}_{T_{u}}(u)\sim\Psi(\mathbb{D}_{H}u^{1-H}),\quad
u\to\infty,$

and the claim follows.

Assume that $H=1/2$. First let (6) holds with $T_{u}=T>0$. We have as
$u\to\infty$ and then $S\to\infty$ (proof is in the Appendix)

(14)
$\displaystyle\mathbb{C}_{T_{u}}(u)\sim\mathbb{P}\left\\{\int\limits_{ut_{*}-S}^{ut_{*}+S}\mathbb{I}\Big{(}B(s)-c_{1}s>q_{1}u,B(s)-c_{2}s>q_{2}u\Big{)}ds>T\right\\}=:\kappa_{S}(u).$

Next with $\phi_{u}$ the density of $B(ut_{*})$,
$\eta=c_{1}t_{*}+q_{1}=c_{2}t_{*}+q_{2}$ and
$\eta_{*}=\eta/t_{*}-c_{2}=q_{2}/t_{*}$ we have

$\displaystyle\kappa_{S}(u)$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{ut_{*}-S}^{ut_{*}}\mathbb{I}(B(s)-c_{2}s>q_{2}u)ds+\\!\\!\\!\\!\int\limits_{ut_{*}}^{ut_{*}+S}\\!\\!\\!\mathbb{I}(B(s)-c_{1}s>q_{1}u)ds>T\big{|}B(ut_{*})=\eta
u-x\right\\}\phi_{u}(\eta u-x)dx$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\Big{\\{}\int\limits_{ut_{*}-S}^{ut_{*}}\mathbb{I}(B(s)-c_{2}s>q_{2}u)ds$
$\displaystyle+\int\limits_{ut_{*}}^{ut_{*}+S}\\!\\!\\!\mathbb{I}(B(s)-B(ut_{*})-c_{1}(s-ut_{*})-c_{1}ut_{*}>q_{1}u-\eta
u+x)ds>T\big{|}B(ut_{*})=\eta u-x\Big{\\}}\phi_{u}(\eta u-x)dx$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{ut_{*}-S}^{ut_{*}}\mathbb{I}(B(s)-c_{2}s>q_{2}u)ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\big{|}B(ut_{*})=\eta
u-x\right\\}\phi_{u}(\eta u-x)dx$ $\displaystyle=$
$\displaystyle\frac{e^{-\frac{\eta^{2}u}{2t_{*}}}}{\sqrt{2\pi
ut_{*}}}\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}Z_{u}(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\}e^{\frac{\eta
x}{t_{*}}-\frac{x^{2}}{2ut_{*}}}dx,$

where $Z_{u}(t)$ is a Gaussian process with expectation and covariance defined
below:

(15) $\displaystyle\mathbb{E}\left\\{Z_{u}(t)\right\\}=\frac{-x}{ut_{*}}t,\ \
\ \ \ \ \operatorname{cov}(Z_{u}(s),Z_{u}(t))=\frac{-st}{ut_{*}}-t,\ \ \ \ \
s\leq t\leq 0.$

Since $Z_{u}(t)$ converges to BM in the sense of convergence finite-
dimensional distributions for any fixed $x\in\mathbb{R}$ as $u\to\infty$ we
have (details are in the Appendix)

$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}Z_{u}(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\}e^{\frac{\eta
x}{t_{*}}-\frac{x^{2}}{2ut_{*}}}dx$ $\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}B(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\}e^{\frac{\eta
x}{t_{*}}}dx$ $\displaystyle=:$ $\displaystyle K(S).$

Since $\mathbb{P}\left\\{\exists t\geq 0:B(t)-ct>x\right\\}=e^{-2cx},\ c,x>0$
(see, e.g., [11]) we have

$\displaystyle K(S)$ $\displaystyle\leq$
$\displaystyle\int\limits_{0}^{\infty}\Big{(}\mathbb{P}\left\\{\exists
s<0:B(s)+\eta_{*}s>x\right\\}+\mathbb{P}\left\\{\exists s\geq
0:B_{*}(s)-c_{1}s>x\right\\}\Big{)}e^{\frac{\eta
x}{t_{*}}}dx+\int\limits_{-\infty}^{0}e^{\frac{\eta x}{t_{*}}}dx$
$\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}\Big{(}e^{(-2\eta_{*}+\eta/t_{*})x}+e^{(-2c_{1}+\eta/t_{*})x}\Big{)}dx+t_{*}/\eta<\infty$

provided by $t_{*}\in(t_{1},t_{2})$. Since $K(S)$ is an increasing function
and $\lim\limits_{S\to\infty}K(S)<\infty$ we have as $S\to\infty$

$\displaystyle K(S)$ $\displaystyle\to$
$\displaystyle\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}B(s)-\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{\infty}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\}e^{\frac{\eta
x}{t_{*}}}dx$ $\displaystyle=$
$\displaystyle\frac{t_{*}}{\eta}\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}\Big{(}B(s)-\frac{\eta_{*}t_{*}}{\eta}s>x\Big{)}ds+\int\limits_{0}^{\infty}\mathbb{I}(B_{*}(s)-\frac{c_{1}t_{*}}{\eta}s>x)ds>\frac{\eta^{2}T}{t_{*}^{2}}\right\\}e^{x}dx$
$\displaystyle=$
$\displaystyle\frac{t_{*}}{\eta}\int\limits_{\mathbb{R}}\mathbb{P}\left\\{\int\limits_{-\infty}^{\infty}\mathbb{I}\Big{(}\sqrt{2}B(s)-|s|+d(s)>x\Big{)}ds>\frac{\eta^{2}T}{2t_{*}^{2}}\right\\}e^{x}dx$
$\displaystyle=$
$\displaystyle\frac{t_{*}}{\eta}\mathcal{B}_{T^{\prime}}^{d}\in(0,\infty),$

where $T^{\prime}$ and $d(s)$ are defined in (10). Finally, combining (3) with
the line above we have as $u\to\infty$ and then $S\to\infty$

$\kappa_{S}(u)\sim\mathcal{B}_{T^{\prime}}^{d}\Psi(\mathbb{D}_{1/2}\sqrt{u})$

and by (14) the claim follows. If (6) holds with $T_{u}=0$, then we obtain the
claim immediately by Theorem 3.1 in [18] and observation that
$\mathcal{B}_{0}^{d}$ coincides with the corresponding Piterbarg constant
introduced in [18].

Now assume that (6) holds with any possible $T_{u}$. If (6) holds with $T>0$,
then for large $u$ and any $\varepsilon>0$ it holds, that
$\mathbb{C}_{(1+\varepsilon)T}(u)\leq\mathbb{C}_{T_{u}}(u)\leq\mathbb{C}_{(1-\varepsilon)T}(u)$
and hence

$(1+o(1))\mathcal{B}_{T^{\prime}(1+\varepsilon)}^{d}\Psi(\mathbb{D}_{1/2}\sqrt{u})\leq\mathbb{C}_{T_{u}}(u)\leq\mathcal{B}_{T^{\prime}(1-\varepsilon)}^{d}\Psi(\mathbb{D}_{1/2}\sqrt{u})(1+o(1)),\qquad
u\to\infty.$

By Lemma 4.1 in [9] $\mathcal{B}_{x}^{d}$ is a continuous function with
respect to $x$ and thus letting $\varepsilon\to 0$ we obtain the claim. If (6)
holds with $T=0$, then for large $u$ and any $\varepsilon>0$ we have

$\mathcal{B}_{\varepsilon}^{d}\Psi(\mathbb{D}_{1/2}\sqrt{u})\leq\mathbb{C}_{T_{u}}(u)\leq\mathcal{B}_{0}^{d}\Psi(\mathbb{D}_{1/2}\sqrt{u})$

and again letting $\varepsilon\to 0$ we obtain the claim by continuity of
$\mathcal{B}_{(\cdot)}^{d}$.

Assume that $H<1/2$. First we have with $\delta_{u}=u^{2H-2}\ln^{2}u$ as
$u\to\infty$ (proof is in Appendix)

(17) $\displaystyle\mathbb{C}_{T_{u}}(u)$ $\displaystyle\sim$
$\displaystyle\mathbb{P}\left\\{\int\limits_{ut_{*}-u\delta_{u}}^{ut_{*}}\mathbb{I}(B_{H}(t)-c_{2}t>q_{2}u)dt>T_{u}\right\\}+\mathbb{P}\left\\{\int\limits_{ut_{*}}^{ut_{*}+u\delta_{u}}\mathbb{I}(B_{H}(t)-c_{1}t>q_{1}u)dt>T_{u}\right\\}$
$\displaystyle=:$ $\displaystyle g_{1}(u)+g_{2}(u).$

Assume that (6) holds with $T>0$. Using the approach from [9] we have with
$\mathbb{I}_{a}(b)=\mathbb{I}(b>a),\ a,b\in\mathbb{R}$

$\displaystyle g_{2}(u)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\delta_{u}T_{u}^{-1}u}\mathbb{I}_{M(u)}\Big{(}\frac{B_{H}(ut_{*}+tT_{u})}{u(q_{1}+c_{1}t_{*})+c_{1}tT_{u}}M(u)\Big{)}dt>1\right\\}$
$\displaystyle=:$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\delta_{u}T_{u}^{-1}u}\mathbb{I}_{M(u)}(Z_{u}^{(1)}(t))dt>1\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\delta_{u}T_{u}^{-1}uK_{1}}\mathbb{I}_{M(u)}(Z_{u}^{(1)}(tK_{1}^{-1}))dt>K_{1}\right\\}$
$\displaystyle=:$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\delta_{u}T_{u}^{-1}uK_{1}}\mathbb{I}_{M(u)}(Z_{u}^{(2)}(t))dt>K_{1}\right\\},$

where

$K_{1}=\frac{T\mathbb{D}_{H}^{1/H}}{2^{\frac{1}{2H}}t_{*}},\quad
M(u)=\inf_{t\in[t_{*},\infty)}\frac{u(c_{1}t+q_{1})}{\text{Var}\\{B_{H}(ut)\\}}=\mathbb{D}_{H}u^{1-H}.$

For variance $\sigma_{Z_{u}^{(2)}}^{2}(t)$ and correlation
$r_{Z_{u}^{(2)}}(s,t)$ of $Z_{u}^{(2)}$ for
$t,s\in[0,\delta_{u}T_{u}^{-1}uK_{1}]$ it holds, that as $u\to\infty$

$\displaystyle 1-\sigma_{Z_{u}^{(2)}}(t)$ $\displaystyle=$
$\displaystyle\frac{2^{\frac{1}{2H}}t_{*}^{H}\mathbb{D}_{H}^{1-1/H}|q_{1}H-(1-H)c_{1}t_{*}|}{(q_{1}+c_{1}t_{*})^{2}}tu^{1-1/H}+O(t^{2}u^{2(1-1/H)}),$
$\displaystyle 1-r_{Z_{u}^{(2)}}(s,t)$ $\displaystyle=$
$\displaystyle\mathbb{D}_{H}^{-2}u^{2H-2}|t-s|^{2H}+O(u^{2H-2}|t-s|^{2H}\delta_{u}).$

Now we apply Theorem 2.1 in [9]. All conditions of the theorem are fulfilled
with parameters

$\displaystyle\omega(x)=x,\ \overleftarrow{\omega}(x)=x,\ \beta=1,\
g(u)=\frac{2^{\frac{1}{2H}}t_{*}^{H}\mathbb{D}_{H}^{1-1/H}|q_{1}H-(1-H)c_{1}t_{*}|}{(q_{1}+c_{1}t_{*})^{2}}u^{1-1/H},$
$\displaystyle\eta_{\varphi}(t)=B_{H}(t),\ \sigma_{\eta}^{2}(t)=t^{2H},\
\Delta(u)=1,\ \varphi=1,$ $\displaystyle n(u)=\mathbb{D}_{H}u^{1-H},\
a_{1}(u)=0,\ a_{2}(u)=\delta_{u}T_{u}^{-1}uK_{1},\ \gamma=0,\ x_{1}=0,\
x_{2}=\infty,\ y_{1}=0,\ y_{2}=\infty,\ x=K_{1},$
$\displaystyle\theta(u)=u^{(1/H-2)(1-H)}\mathbb{D}_{H}^{-1+1/H}|q_{1}H-(1-H)c_{1}t_{*}|^{-1}t_{*}^{H}2^{-\frac{1}{2H}},$

and thus as $u\to\infty$

$g_{2}(u)=\mathbb{P}\left\\{\\!\\!\int\limits_{0}^{\delta_{u}T_{u}^{-1}uK_{1}}\\!\\!\\!\\!\\!\\!\\!\\!\mathbb{I}_{M(u)}(Z_{u}^{(2)}(t))dt>K_{1}\right\\}\sim\mathcal{B}_{2H}(\frac{T\mathbb{D}_{H}^{\frac{1}{H}}}{2^{\frac{1}{2H}}t_{*}})u^{(\frac{1}{H}-2)(1-H)}\frac{t_{*}^{H}\mathbb{D}_{H}^{-1+1/H}}{2^{\frac{1}{2H}}|q_{1}H-(1-H)c_{1}t_{*}|}\Psi(\mathbb{D}_{H}u^{1-H}).$

Similarly we obtain

$g_{1}(u)\sim\mathcal{B}_{2H}(\frac{T\mathbb{D}_{H}^{1/H}}{2^{\frac{1}{2H}}t_{*}})u^{(1/H-2)(1-H)}\frac{t_{*}^{H}\mathbb{D}_{H}^{-1+1/H}}{2^{\frac{1}{2H}}|q_{2}H-(1-H)c_{2}t_{*}|}\Psi(\mathbb{D}_{H}u^{1-H}),\
\ \ u\to\infty$

and the claim follows if in (6) $T>0$. Now let $\eqref{*assumption_T_u}$ holds
with $T=0$. Since
$\mathcal{P}_{T_{u}}(u)\leq\mathbb{C}_{T_{u}}(u)\leq\mathcal{R}_{T_{u}}(u)$ we
obtain the claim by Theorem 2.1 in [16] and Theorem 3.1 in [18]. $\Box$

Proof of Proposition 2.2. The proof of this proposition is the same as the
proof of Proposition 2.2 in [16], thus we refer to [16] for the proof. $\Box$

##  4\. Appendix

Proof of (13). To establish the claim we need to show that

$\displaystyle\mathbb{P}\left\\{\int\limits_{[0,\infty)\backslash[t_{1}-\varepsilon,t_{1}+\varepsilon]}\mathbb{I}\left(V_{1}(s)>u^{1-H}\right)ds>T_{u}/u\right\\}=o(\psi_{1}(T_{u},u)),\
\ u\to\infty.$

Applying Borell-TIS inequality (see, e.g., [24]) we have as $u\to\infty$

$\displaystyle\mathbb{P}\left\\{\int\limits_{[0,\infty)\backslash[t_{1}-\varepsilon,t_{1}+\varepsilon]}\mathbb{I}(V_{1}(s)>u^{1-H})ds>T_{u}/u\right\\}$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\exists
t\in[0,\infty)\backslash[t_{1}-\varepsilon,t_{1}+\varepsilon]:V_{1}(t)>u^{1-H}\right\\}$
$\displaystyle\leq$ $\displaystyle e^{-\frac{(u^{1-H}-M)^{2}}{2m^{2}}},$

where

$M=\mathbb{E}\left\\{\sup\limits_{\exists
t\in[0,\infty)\backslash[t_{1}-\varepsilon,t_{1}+\varepsilon]}V_{1}(t)\right\\}<\infty,\quad
m^{2}=\max\limits_{\exists
t\in[0,\infty)\backslash[t_{1}-\varepsilon,t_{1}+\varepsilon]}\text{Var}\\{V_{1}(t)\\}.$

Since $\text{Var}\\{V_{1}(t)\\}$ achieves its unique maxima at $t_{1}$ we
obtain by (12) that

$e^{-\frac{(u^{1-H}-M)^{2}}{2m^{2}}}=o(\mathbb{P}\left\\{V_{1}(t_{1})>u^{1-H}\right\\}),\quad
u\to\infty$

and the claim follows from the asymptotics of $\psi_{1}(T_{u},u)$ given in
Proposition 3.1. $\Box$

Proof of (14). To prove the claim it is enough to show that as $u\to\infty$
and then $S\to\infty$

$\displaystyle\mathbb{P}\left\\{\int\limits_{[0,\infty)\backslash[ut_{*}-S,ut_{*}+S]}\mathbb{I}\big{(}B(t)-c_{1}t>q_{1}u,B(t)-c_{2}t>q_{2}u\big{)}dt>T\right\\}=o(\mathbb{C}_{T_{u}}(u)),\quad
u\to\infty.$

We have that the probability above does not exceed

$\displaystyle\mathbb{P}\left\\{\exists
t\in[0,\infty)\backslash[ut_{*}-S,ut_{*}+S]:B(t)-c_{1}t>q_{1}u,B(t)-c_{2}t>q_{2}u\right\\}.$

From the proof of Theorem 3.1 in [18], Case (3) and the final asymptotics of
$\mathbb{C}_{T_{u}}(u)$ given in (9) it follows that the expression above
equals $o(\mathbb{C}_{T_{u}}(u))$, as $u\to\infty$ and then $S\to\infty$.
$\Box$

Proof of (3). Define

$G(u,x)=\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}Z_{u}(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}\big{(}B_{*}(s)-c_{1}s>x\big{)}ds>T\right\\}.$

First we show that

(18) $\displaystyle\int\limits_{\mathbb{R}}G(u,x)e^{\frac{\eta
x}{t_{*}}-\frac{x^{2}}{2ut_{*}}}dx=\int\limits_{-M}^{M}G(u,x)e^{\frac{\eta
x}{t_{*}}}dx+A_{M,u},$

where $A_{M,u}\to 0$ as $u\to\infty$ and then $M\to\infty$. We have

$\displaystyle|A_{M,u}|$ $\displaystyle=$
$\displaystyle|\int\limits_{\mathbb{R}}G(u,x)e^{\frac{\eta
x}{t_{*}}-\frac{x^{2}}{2ut_{*}}}dx-\int\limits_{-M}^{M}G(u,x)e^{\frac{\eta
x}{t_{*}}}dx|$ $\displaystyle\leq$
$\displaystyle|\int\limits_{-M}^{M}G(u,x)(e^{\frac{\eta
x}{t_{*}}-\frac{x^{2}}{2ut_{*}}}-e^{\frac{\eta
x}{t_{*}}})dx|+\int\limits_{|x|>M}G(u,x)e^{\frac{\eta
x}{t_{*}}}dx=:|I_{1}|+I_{2}.$

Since the variance of $Z_{u}$ (see (15)) converges to those of BM we have by
Borell-TIS inequality for $x>0$, large $u$ and some $C>0$

(19) $\displaystyle G(u,x)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left\\{\exists
t\in[-S,0):(Z_{u}(t)+\eta_{*}t)>x\right\\}+\mathbb{P}\left\\{\exists
t\in[0,S]:(B_{*}(t)-c_{1}t)>x\right\\}$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left\\{\exists
t\in[-S,0]:(Z_{u}(t)-\mathbb{E}\left\\{Z_{u}(t)\right\\})>x\right\\}+\mathbb{P}\left\\{\exists
t\in[0,S]:B_{*}(t)>x\right\\}\leq e^{-x^{2}/C}.$

Let $u>M^{4}$. For $x\in[-M,M]$ it holds, that
$1-e^{-\frac{x^{2}}{2ut_{*}}}\leq\frac{x^{2}}{2ut_{*}}\leq\frac{1}{M}$ and
hence for $u>M^{4}$ by (19) we have as $M\to\infty$

$\displaystyle|I_{1}|$ $\displaystyle\leq$
$\displaystyle\int\limits_{-M}^{0}e^{\frac{\eta
x}{t_{*}}}(1-e^{-\frac{x^{2}}{2ut_{*}}})dx+\int\limits_{0}^{M}e^{-x^{2}/C+\frac{\eta
x}{t_{*}}}(1-e^{-\frac{x^{2}}{2ut_{*}}})dx\leq\frac{1}{M}\Big{(}\int\limits_{-\infty}^{0}e^{\frac{\eta
x}{t_{*}}}+\int\limits_{0}^{\infty}e^{-x^{2}/C+\frac{\eta x}{t_{*}}}\Big{)}\to
0.$

For $I_{2}$ we have

$\displaystyle I_{2}\leq\int\limits_{-\infty}^{-M}e^{\frac{\eta
x}{t_{*}}}dx+\int\limits_{M}^{\infty}e^{-x^{2}/C}e^{\frac{\eta x}{t_{*}}}dx\to
0,\ \ \ M\to\infty,$

hence (18) holds. Next we show that

$\displaystyle G(u,x)$ $\displaystyle\to$
$\displaystyle\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}B(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\},\
\ \ u\to\infty$

that is equivalent with

$\displaystyle\lim\limits_{u\to\infty}\mathbb{P}\left\\{\int\limits_{-S}^{S}\mathbb{I}\Big{(}X_{u}(s)>x\Big{)}ds>T\right\\}=\mathbb{P}\left\\{\int\limits_{-S}^{S}\mathbb{I}\Big{(}B(s)+k(s)>x\Big{)}ds>T\right\\},$

where $k(s)=\mathbb{I}(s<0)\eta_{*}s-\mathbb{I}(s\geq 0)c_{1}s$ and

$\displaystyle
X_{u}(t)=(Z_{u}(t)+\eta_{*}t)\mathbb{I}(t<0)+(B_{*}(t)-c_{1}t)\mathbb{I}(t\geq
0).$

We have for large $u$

$\displaystyle\mathbb{E}\left\\{(X_{u}(t)-X_{u}(s))^{2}\right\\}=\begin{cases}|t-s|+|t-s|^{2}&t,s\geq
0\\\
-\frac{(s-t)^{2}}{ut_{*}}+|t-s|+\frac{x^{2}(t-s)^{2}}{u^{2}t_{*}^{2}}-\frac{2x(t-s)^{2}\eta_{*}}{ut_{*}}+\eta_{*}^{2}(t-s)^{2}&t,s\leq
0\\\
|t-s|-\frac{s^{2}}{ut_{*}}+\frac{x^{2}s^{2}}{u^{2}t_{*}^{2}}-\frac{2xs(\eta_{*}s+c_{1}t)}{ut_{*}}+(\eta_{*}s+c_{1}t)^{2}&s<0<t\end{cases}$

implying for all $u$ large enough, some $C>0$ and $t,s\in[-S,S+T]$ that

$\displaystyle\mathbb{E}\left\\{(X_{u}(t)-X_{u}(s))^{2}\right\\}\leq C|t-s|.$

Next, by Proposition 9.2.4 in [24] the family $X_{u}(t),\ u>0,\ t\in[-S,S+T]$
is tight in $\mathcal{B}(C([-S,S+T]))$ (Borell $\sigma$-algebra in the space
of the continuous functions on $[-S,S+T]$ generated by the cylindric sets).
As follows from (15), $Z_{u}(t)$ converges to $B(t)$ in the sense of
convergence finite-dimensional distributions as $u\to\infty$, $t\in[-S,S+T]$.
Thus, by Theorems 4 and 5 in Chapter 5 in [2] the tightness and convergence of
finite-dimensional distributions imply weak convergence

$X_{u}(t)\Rightarrow B(t)+k(t)=:W(t),\ \ t\in[-S,S+T].$

By Theorem 11 (Skorohod), Chapter 5 in [2] there exists a probability space
$\Omega$, where all random processes have the same distributions, while weak
convergence becomes convergence almost sure. Thus, we assume that $X_{u}(t)\to
W(t)$ a.s. as $u\to\infty$ as elements of $C[-S,S]$ space with the uniform
metric. We prove that for all $x\in\mathbb{R}$

(20)
$\displaystyle\mathbb{P}\left\\{\lim_{u\to\infty}\int\limits_{-S}^{S}\mathbb{I}(X_{u}(t)>x)dt=\int\limits_{-S}^{S}\mathbb{I}(W(t)>x)dt\right\\}=1.$

Fix $x\in\mathbb{R}$. We shall show that as $u\to\infty$ with probability 1

(21)
$\displaystyle\mu_{\Lambda}\\{t\in[-S,S]\\!:\\!X_{u}(\omega,t)\\!>\\!x\\!>\\!W(\omega,t)\\}\\!+\\!\mu_{\Lambda}\\{t\in[-S,S]\\!:\\!W(\omega,t)\\!>\\!x\\!>\\!X_{u}(\omega,t)\\}\to
0,$

where $\mu_{\Lambda}$ is the Lebesgue measure. Since for any fixed
$\varepsilon>0$ for large $u$ and $t\in[-S,S]$ with probability one
$|W(t)-X_{u}(t)|<\varepsilon$ we have that

$\displaystyle\mu_{\Lambda}\\{t\in[-S,S]:X_{u}(\omega,t)>x>W(\omega,t)\\}+\mu_{\Lambda}\\{t\in[-S,S]:W(\omega,t)>x>X_{u}(\omega,t)\\}$
$\displaystyle\leq$
$\displaystyle\mu_{\Lambda}\\{t\in[-S,S]:W(\omega,t)\in[-\varepsilon+x,\varepsilon+x]\\}.$

Thus, (21) holds if

(22) $\displaystyle\mathbb{P}\left\\{\lim\limits_{\varepsilon\to
0}\mu_{\Lambda}\\{t\in[-S,S]:W(t)\in[-\varepsilon+x,x+\varepsilon]\\}=0\right\\}=1.$

Consider the subset $\Omega_{*}\subset\Omega$ consisting of all $\omega_{*}$
such that

$\lim\limits_{\varepsilon\to
0}\mu_{\Lambda}\\{t\in[-S,S]:W(\omega_{*},t)\in[-\varepsilon+x,x+\varepsilon]\\}>0.$

Then for each $\omega_{*}$ there exists the set
$\mathcal{A}(\omega_{*})\subset[-S,S]$ such that
$\mu_{\Lambda}\\{\mathcal{A(\omega_{*})}\\}>0$ and for
$t\in\mathcal{A(\omega_{*})}$ it holds, that $W(\omega_{*},t)=x$. Thus,

$\mathbb{P}\left\\{\Omega_{*}\right\\}=\mathbb{P}\left\\{\mu_{\Lambda}\\{t\in[-S,S]:W(t)=x\\}>0\right\\},$

the right side of the equation above equals 0 by Lemma 4.2 in [20]. Hence we
conclude that (22) holds, consequently (21) and (20) are true. Since
convergence almost sure implies convergence in distribution we have by (20)
that for any fixed $x\in\mathbb{R}$

$\displaystyle\lim\limits_{u\to\infty}\mathbb{P}\left\\{\int\limits_{-S}^{S}\mathbb{I}(X_{u}(t)>x)dt>T\right\\}=\mathbb{P}\left\\{\int\limits_{-S}^{S}\mathbb{I}(W(t)>x)dt>T\right\\}.$

By the dominated convergence theorem we obtain

$\int\limits_{-M}^{M}G(u,x)e^{\frac{\eta
x}{t_{*}}}dx\to\int\limits_{-M}^{M}\mathbb{P}\left\\{\int\limits_{-S}^{0}\mathbb{I}\Big{(}B(s)+\eta_{*}s>x\Big{)}ds+\int\limits_{0}^{S}\mathbb{I}(B_{*}(s)-c_{1}s>x)ds>T\right\\}e^{\frac{\eta
x}{t_{*}}}dx,\ \ \ \ u\to\infty.$

Thus, the claim follows from the line above and (18). $\Box$

Proof of (17). We have by the proof of Theorem 3.1 in [18], Case (3) and the
final asymptotics of $\mathbb{C}_{T_{u}}(u)$ given in (9)

$\displaystyle\mathbb{P}\left\\{\int\limits_{[0,\infty)\backslash[ut_{*}-u\delta_{u},ut_{*}+u\delta_{u}]}\mathbb{I}(B_{H}(t)-c_{1}t>q_{1}u,B_{H}(t)-c_{2}t>q_{2}u)dt>T_{u}\right\\}$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\exists
t\in[0,\infty)\backslash[ut_{*}-u\delta_{u},ut_{*}+u\delta_{u}]:B_{H}(t)-c_{1}t>q_{1}u,B_{H}(t)-c_{2}t>q_{2}u\right\\}$
$\displaystyle=$ $\displaystyle o(\mathbb{C}_{T_{u}}(u)),\quad u\to\infty$

and hence

$\displaystyle\mathbb{P}\left\\{\int\limits_{[ut_{*}-u\delta_{u},ut_{*}+u\delta_{u}]}\mathbb{I}(B_{H}(t)-c_{1}t>q_{1}u,B_{H}(t)-c_{2}t>q_{2}u)dt>T_{u}\right\\}\sim\mathbb{C}_{T_{u}}(u),\quad
u\to\infty.$

The last probability above is equivalent with $g_{1}(u)+g_{2}(u)$ as
$u\to\infty$, this observation follows from the application of the double-sum
method, see the proofs of Theorem 3.1, Case (3) $H<1/2$ in [18] and Theorem
2.1 in [9] case i). $\Box$

Proof of Proposition 3.1. If $H=1/2$, then an equality takes place, see [9],
Eq. $[5]$. Assume from now on that $H\neq 1/2$. First let (6) holds with
$T>0$. We have for $c>0$ with
$\widetilde{M}(u)=u^{1-H}\frac{c^{H}}{(1-H)^{1-H}H^{H}}$ (recall,
$\mathbb{I}_{a}(b)=\mathbb{I}(b>a),\ a,b\in\mathbb{R}$)

$\displaystyle h_{T_{u}}(u)$ $\displaystyle:=$
$\displaystyle\mathbb{P}\left\\{\int\limits_{0}^{\infty}\mathbb{I}(B_{H}(t)\\!-\\!ct>u)dt>T_{u}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left\\{u(u^{\frac{1}{H}-2}\frac{c^{2}(1-H)^{2-\frac{1}{H}}}{2^{\frac{1}{2H}}H^{2}})\int\limits_{0}^{\infty}\mathbb{I}_{\widetilde{M}(u)}(\frac{B_{H}(tu)\widetilde{M}(u)}{u(1+ct)})dt>T\frac{c^{2}(1-H)^{2-\frac{1}{H}}}{2^{\frac{1}{2H}}H^{2}}\right\\}.$

Next we apply Theorem 3.1 in [9] to calculate the asymptotics of the last
probability above as $u\to\infty$. For the parameters in the notation therein
we have

$\displaystyle\alpha_{0}=\alpha_{\infty}=H,\ \sigma(t)=t^{H},\
\overleftarrow{\sigma}(t)=t^{\frac{1}{H}},\ t^{*}=\frac{H}{c(1-H)},\
A=\frac{c^{H}}{H^{H}(1-H)^{1-H}},\
x=T\frac{c^{2}(1-H)^{2-\frac{1}{H}}}{2^{\frac{1}{2H}}H^{2}}$ $\displaystyle
B=\frac{c^{2+H}(1-H)^{2+H}}{H^{H+1}},\
M(u)=u^{1-H}\frac{c^{H}}{(1-H)^{1-H}H^{H}},\
v(u)=u^{\frac{1}{H}-2}\frac{c^{2}(1-H)^{2-\frac{1}{H}}}{2^{\frac{1}{2H}}H^{2}}.$

and hence we obtain

(23) $\displaystyle h_{T_{u}}(u)\sim
K_{H}\mathcal{B}_{2H}(TD)(C_{H}u^{1-H})^{\frac{1}{H}-1}\Psi(C_{H}u^{1-H}),\ \
\ u\to\infty,$

where

$C_{H}=\frac{c^{H}}{H^{H}(1-H)^{1-H}}\quad\text{ and }\quad
D=2^{-\frac{1}{2H}}c^{2}H^{-2}(1-H)^{2-1/H}.$

Assume that (6) holds with $T=0$. For $\varepsilon>0$ for all large $u$ we
have $h_{\varepsilon u^{1/H-2}}(u)\leq h_{T_{u}}(u)\leq h_{0}(u)$ and thus

$\displaystyle K_{H}\mathcal{B}_{2H}(\varepsilon
D)(C_{H}u^{1-H})^{\frac{1}{H}-1}\Psi(C_{H}u^{1-H})\leq h_{T_{u}}(u)\leq
K_{H}\mathcal{B}_{2H}(0)(C_{H}u^{1-H})^{\frac{1}{H}-1}.$

Since $\mathcal{B}_{2H}(\cdot)$ is a continuous function (Lemma 4.1 in [9])
letting $\varepsilon\to 0$ we obtain (23) for any $T_{u}$ satisfying (6).
Replacing in (23) $u$ and $c$ by $q_{1}u$ and $c_{1}$ we obtain the claim.
$\Box$

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# Nonclassical properties of a particle in a finite range trap: the
$f$-deformed quantum oscillator approach

M. Davoudi Darareh<EMAIL_ADDRESS>M. Bagheri Harouni
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, University
of Isfahan, Hezar Jerib, Isfahan, 81746-73441, Iran

###### Abstract

A particle bounded in a potential with finite range is described by using an
$f$-deformed quantum oscillator approach. Finite range of this potential can
be considered as a controllable deformation parameter. The non-classical
quantum statistical properties of this deformed oscillator can be manipulated
by nonlinearities associated to the finite range.

###### keywords:

Modified Pöschl-Teller like coherent state , nonclassical property ,
$f$-deformed quantum oscillator

###### PACS:

03.65.Fd, 03.65.Ge, 42.50.Dv, 42.50.Ar

##  1 Introduction

The quantum harmonic oscillator, its associated coherent states and their
generalizations [1] play an important role in various theoretical and
experimental fields of modern physics, including quantum optics and atom
optics. Motivations for these generalizations have arisen from symmetry
considerations [2], dynamics [3] and algebraic aspects [4, 5].
The quantum groups approach [4] for generalizing the notion of quantum
harmonic oscillator and its realizations in physical systems, by providing an
algebraic method, has given the possibility of extending the creation and
annihilation operators of the usual quantum oscillator to introduce the
deformed oscillator. In a very general important case, the associated algebra
of this deformed oscillator may be viewed as a deformation of classical Lie
algebra by a generic function $f$, the so-called $f$-deformation function,
depending nonlinearly on the number of excitation quanta and some deformation
parameters. The corresponding oscillator is called an $f$-deformed oscillator
[6]. In contrast to the usual quantum harmonic oscillator, $f$-deformed
oscillators do not have equally-spaced energy spectrum. Furthermore, it has
been known that the most of nonlinear generalizations of some physical models,
such as considered in [7], are only particular cases of $f$-deformed models.
Thus, it is reasonable that $f$-deformed oscillators exhibit strongly various
nonclassical properties [6, 8, 9], such as the sub-Poissonian statistics,
squeezing and the quantum interference effects, displaying the striking
consequences of the superposition principle of quantum mechanics. In addition,
$f$-deformed models depend on one or more deformation parameters which should
permit more flexibility and more ability for manipulating the model [10, 11].
An important question in the $f$-deformed model is the physical meaning of its
deformation parameters. The $q$-deformed oscillator [5], as a special kind of
$f$-deformed oscillators with only one deformation parameter $q$, has been
extensively applied in describing physical models, such as vibrational and
rotational spectra of molecules [12]. The appearance of various nonclassical
features induced by a $q$-deformation relevant to some specific nonlinearity
is also studied [13].
Based on the above-mentioned considerations, $f$-deformed quantum oscillators
and their associated coherent states, such as $f$-coherent states [6] or
nonlinear coherent states [9], can be appropriately established in attempting
to describe certain physical phenomena where their effects could be modelled
through a deformation on their dynamical algebra with respect to conventional
or usual counterparts. This approach has been accomplished, for instance, in
the study of the stationary states of the center-of-mass motion of an ion in
the harmonic trap [9] and under effects associated with the curvature of
physical space [14], the influence of the spatial confinement on the center-
of-mass motion of an exciton in a quantum dot [15], the influence of atomic
collisions and the finite number of atoms in a Bose-Einstein condensate on
controlled manipulation of the nonclassical properties of radiation field
[10], some nonlinear processes in high intensity photon beam [6], intensity-
dependent atom-field interaction in absence and in presence of nonlinear
quantum dissipation in a micromaser [16] and finally, incorporating the
effects of interactions among the particles in the framework of the
$q$-deformed algebra [17].
It is shown that the trapped systems provide a powerful tool for preparation
and manipulation of nonclassical states [18], quantum computations [19] and
quantum communications [20]. Improved experimental techniques have caused
precise measurements on realistic trapping systems, for example, trapped ion-
laser systems [21], trapped gas of atoms [22] and electron-hole carriers
confined in a quantum well and quantum dot [23]. A study of confined quantum
systems using the Wood-Saxon potential [24] and the $q$-analogue harmonic
oscillator trap [25], are some efforts which can be used to explain some
experimentally observed deviations from the results predicted by calculations
based on the harmonic oscillator model.
A realistic case in any experimental setup is that the dimension of the trap
is finite and the realistic trapping potential is not the harmonic oscillator
potential extending to infinity. Thus, the realistic confining potential
becomes flat near the edges of the trap and can be simulated by the tanh-
shaped potential $V(x)=D\,tanh^{2}(x/\delta)$, so-called the modified (or
hyperbolic) Pöschl-Teller(MPT) potential [26]. The MPT potential presents
discrete (or bound) and continuum (or scattering) states. The dynamical
symmetry algebra associated with the bound part of the spectrum is $su(2)$
algebra [27] while for the complete spectra is $su(1,1)$ algebra [28]. The MPT
potential has been used very widely in many branches of physics, such as, atom
optics [29], molecular physics [30] and nanostructure physics [23].
Constructing coherent states for systems with discrete and continuous spectrum
[31] and for various kinds of confining potentials [32] have become a very
important tool in the study of some quantum systems. The Pöschl-Teller(PT)
potentials, including trigonometric PT(TPT) and MPT potentials with discrete
infinite and finite dimensional bound states respectively, because of their
relations to several other trapping potentials are of crucial importance. Some
types of the coherent states for the MPT potential have been constructed. The
minimum-uncertainty coherent states formalism [33], the Klauder-Perelomov
approach [1] by realization of lowering and raising operators in terms of the
physical variable $u=tanh(x/\delta)$ by means of factorizations [34] and
applying one kind generalized deformed oscillator algebra with a selected
deformed commutation relation [35], are some attempts for this purpose.
In the present paper, we intend to investigate the nonlinear effects appeared
due to finite dimension of the trapping potential on producing new
nonclassical quantum statistical properties using the $f$-deformed quantum
oscillator approach. For this aim, it will be shown that the finite range of
the trapping potential leads to the $f$-deformation of the usual harmonic
potential with the well depth $D$ as a controllable physical deformation
parameter. Then, the $f$-deformed bound coherent states [36] for the above-
mentioned MPT quantum oscillator are introduced and their nonclassical
properties are examined. We think that by this $f$-deformed quantum oscillator
approach the problem of trapped ion-laser system and trapped gas of atoms,
such as a Bose-Einstein condensate, in a realistic trap can be studied
analytically.
The paper is organized as follows. In section 2, we introduce the $f$-deformed
quantum oscillator equivalent to the MPT oscillator and obtain the associated
ladder operators. In section 3, we construct the $f$-deformed bound coherent
states of the MPT quantum oscillator and examine its resolution of identity.
Section 4 devoted to the study of the influence of the finite range potential
on producing and manipulating the nonclassical properties, including the sub-
Poissonian statistics and squeezing character. Finally, the summary and
conclusions are presented in section 5.

##  2 MPT Hamiltonian as an $f$-deformed quantum oscillator

In this section, we will consider a bounded particle inside the MPT potential,
called the MPT oscillator, and we will associate to this system an
$f$-deformed quantum oscillator. By using this mathematical model, we try to
investigate physical deformation parameters in the model, to manipulate the
nonlinearities related to the finite range effects on this system. For this
purpose, we first give the bound energy eigenvalues for the MPT potential.
Then, by comparing it with the energy spectrum of the general $f$-deformed
quantum oscillator, we will obtain the deformed annihilation and creation
operators.
Let us consider the MPT potential energy

$V(x)=D\,tanh^{2}(\frac{x}{\delta}),$ (1)

where $D$ is the depth of the well, $\delta$ determines the range of the
potential and $x$ gives the relative distance from the equilibrium position.
The well depth, D, can be defined as $D=\frac{1}{2}m\omega^{2}\delta^{2}$,
with mass of the particle $m$ and angular frequency $\omega$ of the harmonic
oscillator, so that, in the limiting case $D\rightarrow\infty$(or
$\delta\rightarrow\infty)$, but keeping the product $m\omega^{2}$ finite, the
MPT potential energy reduces to harmonic potential energy,
$\lim_{D\rightarrow\infty}V(x)=\frac{1}{2}m\omega^{2}x^{2}$. Figure 1 depicts
the MPT potential for three different values of the well depth $D$. Harmonic
potential limit by increasing $D$ is clear from this figure. Solving the
Schrödinger equation, the energy eigenvalues for the MPT potential are
obtained as [37]

$E_{n}=D-\frac{\hbar^{2}\omega^{2}}{4D}(s-n)^{2},\quad\quad
n=0,1,2,\cdots,[s]$ (2)

in which $s=(\sqrt{1+(\frac{4D}{\hbar\omega})^{2}}-1)/2$, and $[s]$ stands for
the closest integer to $s$ that is smaller than $s$. The MPT oscillator
quantum number $n$ can not be larger than the maximum number of bound states
$[s]$, because of the dissociation condition $s-n\geq 0$. Consequently, the
total number of bound states is $[s]+1$. We should note that for integer $s$,
the final bound state and the total number of bound states will be $s-1$ and
$s$, respectively. Also, for every small value of the well depth $D$, we
always have at least one bound state for the MPT oscillator, i.e., the ground
state. By introducing a dimensionless parameter
$N=\frac{4D}{\hbar\omega}=\frac{2m\omega\delta^{2}}{\hbar}$, the total number
of bound states will obtain from $[(\sqrt{1+(N)^{2}}-1)/2]+1$. For integer
$s$, a simple relation $N=2\sqrt{s(s+1)}$ will connect $N$ to the total number
of bound states, i.e., $s$. The bound energy spectrum in equation (2) can be
rewritten as

$E_{n}=\hbar\omega[-\frac{n^{2}}{N}+(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})n+\frac{1}{2}(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})].$
(3)

The relation (3) shows a nonlinear dependence on the quantum number $n$, so
that, different energy levels are not equally spaced. It is clear that, in the
limit $D\rightarrow\infty$ (or $N\rightarrow\infty$), the energy spectrum for
the quantum harmonic oscillator will be obtained, i.e.,
$E_{n}=\hbar\omega(n+\frac{1}{2})$. In contrast with some confined systems
such as a particle bounded in an infinite and finite square well potentials,
by decreasing the size of the confinement parameter, i.e., the finite range
$\delta$ of the MPT oscillator, energy eigenvalues decreases.
A quantity that has a close connection to experimental information is the
energy level spacing, $E_{n+1}-E_{n}$, where it corresponds to the transition
frequency between two adjacent energy levels. Furthermore, by this quantity
one can theoretically explore an algebraic representation for the quantum
mechanical potentials with discrete spectrum [38]. Based upon above
considerations, a useful illustration for the effects of the deformation
parameter $D$ on the nonlinear behavior of the deformed oscillator, can be
investigated by introducing the delta parameter $\Delta_{n}$ as

$\Delta_{n}=\frac{E_{n+1}-E_{n}}{\hbar\omega}-1$ (4)

which measures the amount of deviation of the adjacent energy level spacing of
the deformed oscillator with respect to the non-deformed or harmonic
oscillator. Substituting from equation (3) in equation (4) we can ontain the
delta parameter $\delta$ for the MPT potential

$\Delta_{n}=-\frac{2}{N}n+\sqrt{1+\frac{1}{N^{2}}}-\frac{2}{N}-1$ (5)

On the other hand, the $f$-deformed quantum oscillator [6], as a nonlinear
oscillator with a specific kind of nonlinearity, is characterized by the
following deformed dynamical variables $\hat{A}$ and $\hat{A}^{\dagger}$

$\displaystyle\hat{A}$ $\displaystyle=$
$\displaystyle\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a},$
$\displaystyle\hat{A}^{\dagger}$ $\displaystyle=$ $\displaystyle
f(\hat{n})\hat{a}^{\dagger}=\hat{a}^{\dagger}f(\hat{n}+1),\quad\quad\hat{n}=\hat{a}^{\dagger}\hat{a},$
(6)

where $\hat{a}$ and $\hat{a}^{\dagger}$ are usual boson annihilation and
creation operators $([\hat{a},\hat{a}^{\dagger}]=1)$, respectively. The real
deformation function $f(\hat{n})$ is a nonlinear operator-valued function of
the harmonic number operator $\hat{n}$, where it introduces some
nonlinearities to the system. From equation (2), it follows that the
$f$-deformed operators $\hat{A}$, $\hat{A}^{\dagger}$ and $\hat{n}$ satisfy
the following closed algebra

$\displaystyle[\hat{A},\hat{A}^{\dagger}]=$
$\displaystyle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n}),$ (7)
$\displaystyle[\hat{n},\hat{A}]=$
$\displaystyle-\hat{A},\quad\quad[\hat{n},\hat{A}^{{\dagger}}]=\hat{A}^{{\dagger}}.$

The above-mentioned algebra, represents a deformed Heisenberg-Weyl algebra
whose nature depends on the nonlinear deformation function $f(\hat{n})$. An
$f$-deformed oscillator is a nonlinear system characterized by a Hamiltonian
of the harmonic oscillator form

$\hat{H}=\frac{\hbar\omega}{2}(\hat{A}^{\dagger}\hat{A}+\hat{A}\hat{A}^{\dagger}).$
(8)

Using equation (2) and the number state representation
$\hat{n}|n\rangle=n|n\rangle$, the eigenvalues of the Hamiltonian (8) can be
written as

$E_{n}=\frac{\hbar\omega}{2}[(n+1)f^{2}(n+1)+nf^{2}(n)].$ (9)

It is worth noting that in the limiting case $f(n)\rightarrow 1$, the deformed
algebra (7) and the deformed energy eigenvalues (9) will reduce to the
conventional Heisenberg-Weyl algebra and the harmonic oscillator spectrum,
respectively.
Comparing the bound energy spectrum of the MPT oscillator, equation (3), and
the energy spectrum of an $f$-deformed oscillator, equation (9), we obtain the
corresponding deformation function for the MPT oscillator as

$f^{2}(\hat{n})=\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}.$ (10)

Furthermore, the ladder operators of the bound eigenstates of the MPT
Hamiltonian can be written in terms of the conventional operators $\hat{a}$
and $\hat{a}^{\dagger}$ as follows

$\hat{A}=\hat{a}\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}},\quad\quad\hat{A}^{\dagger}=\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}}\hat{a}^{\dagger}.$
(11)

These two operators satisfy the deformed Heisenberg-Weyl commutation relation

$[\hat{A},\hat{A}^{\dagger}]=\sqrt{1+\frac{1}{N^{2}}}-\frac{2\hat{n}+1}{N},$
(12)

and they act upon the quantum number states $|n\rangle$, corresponding to the
energy eigenvalues $E_{n}$ given in equation (3), as

$\displaystyle\hat{A}|n\rangle$ $\displaystyle=$ $\displaystyle
f(n)\sqrt{n}|n-1\rangle,$ $\displaystyle\hat{A}^{{\dagger}}|n\rangle$
$\displaystyle=$ $\displaystyle f(n+1)\sqrt{n+1}|n+1\rangle.$ (13)

The commutation relation (12), can be identified with the usual $su(2)$
commutation relations by introducing the set of transformations

$\hat{A}\rightarrow\frac{\hat{J}_{+}}{\sqrt{N}},\quad\hat{A}^{{\dagger}}\rightarrow\frac{\hat{J}_{-}}{\sqrt{N}},\quad\hat{n}\rightarrow\frac{\sqrt{1+N^{2}}-1}{2}-\hat{J}_{0},$
(14)

where $\hat{J}_{\mu}$ satisfy the usual angular momentum relations [39]. The
$f$-deformed commutation relation (12) in a special case of large but finite
value of $N$, which corresponds to the small deformation, can lead to a maths-
type $q$-deformed commutation relation [40], i.e.,
$\hat{A}\hat{A}^{{\dagger}}-q\hat{A}^{{\dagger}}\hat{A}=1$, with
$q=1-\frac{2}{N}=1-\frac{\hbar\omega}{2D}$. The harmonic oscillator limit
corresponds to $D\rightarrow\infty$ then $q\rightarrow 1$. This result
confirms a correspondence between the $q$-deformed oscillators and finite
range potentials, which is studied elsewhere [41].
It is evident that, herein, we have focused our attention on the quantum
states of the MPT Hamiltonian which exhibit bound oscillations with finite
range. The remaining states, i.e., the scattering states or energy continuum
eigenstates, have non-evident boundary conditions. From physical point of
view, it means that the excitation energies of this confined system in the MPT
potential energy are small compared with the well depth potential energy $D$,
such that, only the vibrational modes dominated and the scattering or
continuum states should be neglected. Some important physical systems with
such circumstances are vibrational excitations of molecular systems [42],
trapped ions or atoms [43] and the electron-hole carriers confined in a
quantum well [23].

##  3 $f$-Deformed bound coherent states

In the context of the $f$-deformed quantum oscillator approach, we introduce
the $f$-deformed bound coherent states $|\alpha,f\rangle$ for the MPT
oscillator as a coherent superposition of all bound energy eigenstates of the
MPT Hamiltonian as below

$|\alpha,f\rangle=C_{f}\sum_{n=0}^{[s]}\frac{\alpha^{n}}{\sqrt{n!}f(n)!}|n\rangle,\quad
C_{f}=\left(\sum_{n=0}^{[s]}\frac{|\alpha|^{2n}}{n!(f(n)!)^{2}}\right)^{-1/2},$
(15)

so that $\hat{n}|n\rangle=n|n\rangle$, and $f(n)!=f(n)f(n-1)\cdots f(0)$,
where $f(n)$ is obtained in equation (10). Since the sum in the equation (15)
is finite, the states $|\alpha,f\rangle$, similar to the Klauder-Perelomov
coherent states [1], are not an eigenstate of the annihilation operator
$\hat{A}$. From equations (2) and (15), we arrive at

$\hat{A}|\alpha,f\rangle=\alpha|\alpha,f\rangle-\frac{C_{f}\alpha^{[s]+1}}{\sqrt{[s]!}f([s])!}|[s]\rangle.$
(16)

As is clear from this equation, these states can not be considered as a right-
hand eigenstate of annihilation operator $\hat{A}$. This property is common
character of all coherent states that are defined in a finite- dimensional
basis [36, 44].
The ensemble of the $f$-deformed bound coherent states $|\alpha,f\rangle$
labelled by the complex number $\alpha$ form an overcomplete set with the
resolution of the identity

$\int d^{2}\alpha|\alpha,f\rangle
m_{f}(|\alpha|)\langle\alpha,f|=\sum_{n=0}^{[s]}|n\rangle\langle
n|=\hat{\textbf{1}},$ (17)

where $m_{f}(|\alpha|)$ is the proper measure for this family of the bound
coherent states. Substituting from equation (15) in equation (17) and using
integral relation
$\int_{0}^{\infty}K_{\nu}(t)t^{\mu-1}dt=2^{\mu-2}\Gamma(\frac{\mu-\nu}{2})\Gamma(\frac{\mu+\nu}{2})$
for the modified Bessel function $K_{\nu}(t)$ of the second kind and of the
order $\nu$, we obtain the suitable choice for the measure function as

$m_{f}(|\alpha|)=\frac{K_{\nu}(|\alpha|)}{2^{l}\pi|\alpha|^{\nu}C_{f}^{2}(|\alpha|)},$
(18)

where $\nu=(1+\gamma)n-\eta$, $l=(1-\gamma)n+\eta+1$ and $\gamma=\frac{1}{N}$,
$\eta=\sqrt{1+\frac{1}{N^{2}}}$.

In contrast to the Gazeau-Klauder coherent states [31], the $f$-deformed
coherent states, such as introduced in equation (15), do not generally have
the temporal stability [6]. But it is possible to introduce a notion of
temporally stable $f$-deformed coherent states [45].

##  4 Quantum statistical properties of the MPT oscillator

###  4.1 Sub-Poissonian statistics

In order to determine the quantum statistics of the MPT quantum oscillator, we
consider Mandel parameter $Q$ defined by [46]

$Q=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}}{\langle\hat{n}\rangle}-1.$
(19)

The sub-Poissonian statistics (antibunching effect), as an important
nonclassical property, exists whenever $Q<0$. When $Q>0$, the state of the
system is called super-Poissonian (bunching effect). The state with $Q=0$ is
called Poissonian. Calculating the Mandel parameter $Q$ in equation (19) over
the $f$-deformed bound coherent states $|\alpha,f\rangle$ defined in equation
(15), it can be described the finite range dependence of the Mandel parameter.
Figure 2 shows the parameter $Q$ for four different values of $|\alpha|$,
i.e., $|\alpha|=3,\,4,\,5,\,7$. As is seen, for every one of the values of
$|\alpha|$, the Mandel parameter $Q$ exhibits the sub-Poissonian statistics at
certain range of $D$ or the dimensionless parameter
$N=\frac{4D}{\hbar\omega}$, where this range is determined by the value of
$|\alpha|$. The bigger parameter $|\alpha|$ is, the more late the Mandel
parameter tends to the Poissonian statistics. As expected, with further
increasing values of $D$ or $N$, the Mandel parameter $Q$ finally stabilized
at an asymptotical zero value, corresponding to the Poissonian statistics
associated to the canonical harmonic oscillator coherent states. For the limit
$N\rightarrow 0$(or $D\rightarrow 0$) and for every values of $|\alpha|$, the
Mandel parameter becomes $Q=-1$, where it is reasonable, because in this
limit, only the ground state supports by the potential.

###  4.2 Quadrature squeezing

As another important nonclassical property, we examine the quadrature
squeezing of the MPT quantum oscillator. For this purpose, we consider
quadrature operators $\hat{q}_{\varphi}$ and $\hat{p}_{\varphi}$ defined as
[47]

$\hat{q}_{\varphi}=\frac{1}{\sqrt{2}}(\hat{a}e^{-i\varphi}+\hat{a}^{\dagger}e^{i\varphi}),\quad\quad\hat{p}_{\varphi}=\frac{i}{\sqrt{2}}(\hat{a}^{{\dagger}}e^{i\varphi}-\hat{a}e^{-i\varphi}),$
(20)

satisfying the commutation relation $[\hat{q}_{\varphi},\hat{p}_{\varphi}]=i$.
One can define the invariant squeezing coefficient $S$ as the difference
between the minimal value (with respect to the phase $\varphi$) of the
variances of each quadratures and the mean value $1/2$ of these variances in
the coherent or vacuum state. Simple calculations result in the formula

$S=\langle\hat{a}^{{\dagger}}\hat{a}\rangle-|\langle\hat{a}\rangle|^{2}-|\langle\hat{a}^{2}\rangle-\langle\hat{a}\rangle^{2}|,$
(21)

so that the condition of squeezing is $S<0$. Calculating the squeezing
parameter $S$ over the $f$-deformed bound coherent states in equation (15), we
examine the squeezed character of these states. In figure 3, we have plotted
the parameter $S$ with respect to the dimensionless deformation parameter
$N=\frac{4D}{\hbar\omega}$ for three different values of $|\alpha|$, namely
$|\alpha|=0.5,\,1,\,1.3$. As is seen, the states $|\alpha,f\rangle$ exhibit
squeezing for certain values of $|\alpha|$. Furthermore, the squeezing
character of the states $|\alpha,f\rangle$ tend to zero as $N$ or the well
depth $D$ of the MPT potential approaches to infinity, according to the
coherent states of the quantum harmonic oscillator. In the limit $N\rightarrow
0$(or $D\rightarrow 0$), this plot shows the quadrature squeezing $S=0$, where
it is in agreement with the only ground state supported by the potential in
this limit.

##  5 Conlusions

In this paper, we have introduced an algebraic approach based on the
$f$-deformed quantum oscillator for considering a particle in the real
confining potential which has finite trap dimension, in contrast to the
harmonic oscillator potential extending to infinity. Proposed confining model
potential is the modified Pöschl-Teller potential. We have shown that the
effects of the finite trap dimension in this model potential can be considered
as a natural deformation in the quantum harmonic oscillator algebra. This
quantum deformation approach makes possible analytical study of a wide
category of realistic bound quantum systems algebraically. It is shown that
the nonlinear behavior resulted from this finite range effects can lead to
generate and manipulate some important nonclassical properties for this
deformed quantum oscillator. We have obtained that the presented $f$-deformed
bound coherent states of the modified Pöschl-Teller potential can exhibit the
sub-Poissonian statistics and quadrature squeezing in definite domain of the
trap dimension or well depth $D$ of this potential. In the large but finite
value for the well depth $D$, i.e., small deformation, a $q$-deformed
oscillator with $q=1-\hbar\omega/(2D)$ will result. In the limit
$D\rightarrow\infty$, the harmonic oscillator counterpart is obtained.
Based on the approach in this paper, we can obtain exact solutions for
realistic confined physical systems such as, trapped ion-laser system (in
progress), Bose-Einstein condensate and confined carriers in nano-structures.

Acknowledgments
The authors wish to thank The Office of Graduate Studies and Research Vice
President of The University of Isfahan for their support.

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Figure captions

Fig. 1. Plots of the MPT potential for three different values of the well
depth $D$, $D=1$(solid curve), $D=2$(dashed curve) and
$D\rightarrow\infty$(dotted curve).

Fig. 2. Plots of the Mandel parameter $Q$ versus the dimensionless deformation
parameter $N$ for $|\alpha|=3$(solid curve), $|\alpha|=4$(dashed curve),
$|\alpha|=5$(dotted curve) and $|\alpha|=7$(dash-dotted curve).

Fig. 3. Plots of the invariant squeezing coefficient $S$ versus the
dimensionless deformation parameter $N$ for $|\alpha|=0.5$(solid curve),
$|\alpha|=1$(dashed curve) and $|\alpha|=1.3$(dotted curve).