\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 05, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/05\hfil A generalization of Osgood's test]
{A generalization of Osgood's test and a comparison criterion for
integral equations with noise}

\author[M. J. Ceballos-Lira, J. E. Mac\'ias-D\'iaz, J. Villa\hfil EJDE-2011/05\hfilneg]
{Marcos J. Ceballos-Lira, Jorge E. Mac\'ias-D\'iaz,
Jos\'e Villa}  % in alphabetical order

\address{Marcos Josias Ceballos-Lira \newline
Divisi\'on Acad\'emica de Ciencias B\'asicas,
Universidad Ju\'arez Aut\'onoma de Tabasco,\newline
Km. 1 Carretera Cunduac\'an-Jalpa de M\'endez,
Cunduac\'an, Tab. 86690, Mexico}
\email{marjocel\_81@hotmail.com}

 \address{Jorge Eduardo Mac\'ias-D\'iaz \newline
Departamento de Matem\'aticas y F\'{\i}sica,
Universidad Aut\'onoma de Aguascalientes, \newline
Avenida Universidad 940, Ciudad Universitaria,
Aguascalientes, Ags. 20131, Mexico}
\email{jemacias@correo.uaa.mx}

\address{Jos\'e Villa Morales\newline
Departamento de Matem\'aticas y F\'{\i}sica,
Universidad Aut\'onoma de Aguascalientes, \newline
Avenida Universidad 940, Ciudad Universitaria,
Aguascalientes, Ags. 20131, Mexico}
\email{jvilla@correo.uaa.mx}

\thanks{Submitted December 7, 2010. Published January 12, 2011.}
\subjclass[2000]{45G10, 45R05, 92F05, 74R10, 74R15}
\keywords{Osgood's test; comparison criterion; time of explosion;
\hfill\break\indent integral equations with noise; crack failure}

\begin{abstract}
 In this article, we prove a generalization of Osgood's test
 for the explosion of the solutions of initial-value problems.
 We also establish a comparison criterion for the solution of
 integral equations with noise, and provide estimations of
 the time of explosion of problems arising in the investigation
 of crack failures where the noise is the absolute value of the
 Brownian motion.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction\label{S:Intro}}

Let $x_0$ be a positive, real number, let $b$ be a positive,
real-valued function defined on $[0,\infty )$, and suppose that
$y$ is an extended real-valued function with the same domain as
$b$. The present work is motivated by a criterion for the
explosion of the solutions of ordinary differential equations of
the form
\begin{equation}
\begin{gathered}
\frac{dy(t)}{dt}=b(y(t)), \quad t>0, \\
y(0)=x_0.
\end{gathered} \label{Eq:ODE}
\end{equation}
More precisely, the time of explosion of the solution of this
initial-value problem is the nonnegative, extended real number
$t_e=\sup \{t\geq 0:y(t)<\infty \}$. The above-mentioned
criterion is called \emph{Osgood's test} after its author
\cite{Osgood}, and it states that $t_e$ is finite if and only if
$\int_{x_0}^{\infty }ds/b(s)<\infty $. In such case,
$t_e=\int_{x_0}^{\infty }ds/b(s)$.

A natural question readily arises about the possibility to extend
Osgood's test to more general, initial-value problems, say, to
problems in which the drift function $b$ in the ordinary
differential equation of \eqref{Eq:ODE} is multiplied by a
suitable, nonnegative function of $t$. Another direction of
investigation would be to investigate conditions under which the
solutions of the integral form of such equation with a noise
function added, explode in a finite time. Evidently the
consideration of these two problems as a single one is an
interesting topic of study \emph{per se}. In fact, the purpose of
this paper is to provide a generalization of Osgood's test to
integral equations with noise, which generalize the problem
presented in \eqref{Eq:ODE}. Important, as it is in the recent
literature \cite{Kafini, L-V}, the problem of establishing
analytical conditions under which the time of explosion of the
problem under investigation is finite, is tackled here. In the way, we
establish a comparison criterion for the solutions of integral
equations with noise, and show some applications to the spread of
cracks in rigid surfaces.

Our manuscript is divided in the following way: Section
\ref{S:Osgood} introduces the integral equation with noise that
motivates this manuscript, along with a convenient simplification
for its study; a generalization of Osgood's test is presented in
this stage for  the associated initial-value problem for both
scenarios: noiseless and noisy systems. Section \ref{S:Compar}
establishes a comparison criterion for the solutions of two
noiseless systems with comparable initial conditions. A necessary
condition for the explosion of the solutions of the problem under
investigation is provided in this section, together with an
illustrative counterexample and a partial converse. In Section
\ref{S:Approx}, we give upper and lower bounds for the value of
the time of explosion of our integral equation. Finally, Section
\ref{S:Applic} provides estimates of probabilities associated to
the time of explosion of a system in which the noise is the
absolute value of the Brownian motion.

\section{Osgood's test\label{S:Osgood}}

Let $\overline {\mathbb{R}}$ denote the set of extended real
numbers. Throughout, $a, b : [0 , \infty) \to \mathbb{R}$ will
represent positive, continuous functions, while the function $g :
[0 , \infty) \to \mathbb{R}$ will be continuous and nonnegative.
For physical reasons, the function $g$ is called a \emph{noise}.
In this work, $x _0$ will denote a positive, real number, and
$X :[0 , \infty) \to \overline {\mathbb{R}}$ will be a nonnegative
function whose dependency on $t \geq 0$ is represented by $X _t$.
We are interested in establishing
conditions under which the solutions of the integral equation
\begin{equation}  \label{Eq:Model}
X _t = x _0 + \int _0 ^t a (s) b (X _s) d s + g (t), \quad t \geq 0,
\end{equation}
explode in finite time. More precisely, we define the \emph{time
of explosion} of $X$ as the nonnegative, extended real number
$T^X _e = \sup \{t \geq 0 : X _t < \infty \}$. In this manuscript, we
investigate conditions under which the time of explosion of $X$ is
a real number.

Letting $Y _t = X _t - g (t)$, one sees immediately that the
problem  under consideration is equivalent to finding the time of
explosion of the solution $Y$ of the equation
\begin{equation}  \label{Eq:EIM}
Y _t = x _0 + \int _0 ^t a (s) b (Y _s + g (s)) d s, \quad t \geq 0.
\end{equation}
As a matter of fact, $T _e ^X = T _e ^Y$. From this point on, this
common, extended real number will be denoted simply by $T _e$ for
the sake of briefness.

\begin{remark} \label{rmk1} \rm
It is worth noticing that \eqref{Eq:EIM} can be presented in
differential form as the equivalent, initial-value problem
\begin{equation}
\begin{gathered}
\frac{dY_{t}}{dt}=a(t)b(Y_{t}+g(t)), \quad t>0, \\
Y_0=x_0,
\end{gathered}  \label{Eq:EDM}
\end{equation}
a problem for which the existence of solutions is guaranteed, for
instance, when $b$ is locally Lipschitzian and $a$ is regulated
(see  \cite[(10.4.6)]{Dieudonne})
\end{remark}

Let $r$ be a real number such that $0<r\leq x_0$. We define the
functions $ A:[0,\infty )\to \mathbb{R}$ and
$B_{r}:[x_0-r,\infty )\to \mathbb{R}$ by
\[
A(t)=\int_0^{t}a(s)ds\quad \text{and}\quad
B_{r}(x)=\int_{x_0-r}^x\frac{ds}{b(s)}.
\]
For the sake of convenience, we let $B$ be the function $B_0$.
Evidently, both of these functions are nonnegative,
increasing and continuous, and so
are their inverses. On the other hand, if $r\geq -x_0$,
 we let $\widetilde{B}_{r}:[x_0+r,\infty )\to \mathbb{R}$ be
given by $\widetilde{B}_{r}=B_{-r}$. For every fixed $x\geq x_0$,
we define $\widetilde{B}^x:[-x_0,x-x_0]\to \mathbb{R}$ by
$\widetilde{B}^x(r)= \widetilde{B}_{r}(x)$; we prefer this second
notation in either case.
Additionally, we define $\beta :[-x_0,\infty )\to \mathbb{R}$ by
$\beta (t)=\widetilde{B}^{\infty }(t)$. All of these functions
and their inverses are nonnegative, continuous and decreasing in
their domains.

\begin{lemma}[Generalized Osgood's test]
The initial-value problem
\begin{equation}  \label{Eq:EGO}
\begin{gathered}
 \frac {d y (t)} {d t} = a (t) b (y (t)), \quad t > 0, \\
y (0) = x _0,
\end{gathered}
\end{equation}
has a unique solution given by $y (t) = B ^{- 1} (A (t))$, for
$t < A ^{- 1} (B (\infty))$. The solution explodes in finite
time if and only if $B (\infty) < A (\infty)$, in which case,
$T _e ^y = A ^{- 1} (B (\infty))$.
\end{lemma}

\begin{proof}
The function $y(t)=B^{-1}(A(t))$ is evidently a solution of
\eqref{Eq:EGO}. Additionally, expressing the differential equation
in \eqref{Eq:EGO} as $y'(s)/b(y(s))=a(s)$, integrating both sides
over $[0,t]$ and performing a suitable substitution, we obtain
that $B(y(t))=A(t)$, whence
the uniqueness follows. Moreover, $y(t)$ is real if and only if
$t<A^{-1}(B(\infty ))$.

Now, if the solution of \eqref{Eq:EGO} explodes at the time
$t_e<\infty $,
then $B(\infty )=A(t_e)<A(\infty )$. Conversely, the number
$A^{-1}(B(\infty ))$ is real, so that
\[
B(y(A^{-1}(B(\infty )))) = A(A^{-1}(B(\infty ))) = B(\infty ).
\]
This implies that $T_e^{y}\leq A^{-1}(B(\infty ))$, and the opposite
inequality follows from the fact that the solution of \eqref{Eq:EGO}
exists for $t<A^{-1}(B(\infty ))$.
\end{proof}

As a consequence, the solution of \eqref{Eq:EGO} is nonnegative,
continuous and increasing on $[0,T_e^{y})$, and so is its inverse
on $[x_0,\infty )$. Likewise, the function
$\overline{B}:[x_0,\infty )\to \mathbb{R}$, given by the formula
\[
\overline{B}(y)=\int_{x_0}^{y}\frac{ds}{b(s+g(Y^{-1}(s)))},
\]
is nonnegative, continuous and increasing.

\begin{corollary} \label{Coro:1}
The solution of \eqref{Eq:EDM} can be expressed as $Y_{t}=
\overline{B}^{-1}(A(t))$, for every $t<A^{-1}(\overline{B}(\infty ))$.
\end{corollary}

The proof of the above corollary follows as in Osgood's test.


\section{A comparison theorem\label{S:Compar}}

\begin{theorem}[Comparison criterion]\label{Thm:Comparison}
Let $0<x_0\leq x_1$, let $b$ be
non-de\-crea\-sing, and assume that the functions $u,v:[0,\infty
)\to \overline{\mathbb{R}}$ satisfy
\[
v(t)\geq x_1+\int_0^{t}a(s)b(v(s))ds\quad \text{and}\quad
u(t)=x_0+\int_0^{t}a(s)b(u(s))ds,\quad t\geq 0.
\]
Then, $v(t)\geq u(t)$ for every $t\geq 0$, and $T_e^v\leq
A^{-1}(B(\infty ))$.
\end{theorem}

\begin{proof}
It is sufficient to show that $v\geq u$ because, in such case,
$T_e^v\leq T_e^u=A^{-1}(B(\infty ))$. Assume first that $x_0<x_1$,
and let $N=\{t\geq 0:u(s)\leq v(s),s\in [ 0,t]\}$. The set $N$ is
nonempty, so $\widetilde{T}=\sup N$ exists in $\overline{\mathbb{R}}$.
If $\widetilde{T}$ were a real number, then
\[
L=\lim_{\epsilon \to 0^{+}}(v(\widetilde{T}+\epsilon )-u(\widetilde{T
}+\epsilon ))
\geq x_1-x_0+\lim_{\epsilon \to 0^{+}}\int_{
\widetilde{T}}^{\widetilde{T}+\epsilon }a(s)
[ b(v(s))-b(u(s))] ds
\]
by the fact that $v(s)-u(s)\geq 0$, for every $s\in [ 0,\widetilde{T}]$.
It follows that $L\geq x_1-x_0$. By definition, there exists
$\delta>0 $ such that $v(\widetilde{T}+s)-u(\widetilde{T}+s)>0$
for every $s\in[ 0,\delta )$, whence it follows that
$\widetilde{T}+\frac{\delta }{2}\in N$, a contradiction.
Consequently, $u(t)\leq v(t)$ for every $t\geq 0$.
Now, in case that $x_0=x_1$, the solution of the equation
\[
u_{r}(t)=x_0-r+\int_0^{t}a(s)b(u_{r}(s))ds,\quad 0<r<x_0,
\]
satisfies $v(t)\geq u_{r}(t)$, for every $t\geq 0$.
Using Osgood's test and
the continuity of $B_{r}^{-1}$, we obtain
\[
v(t)\geq \lim_{r\to 0^{+}}u_{r}(t)=\lim_{r\to
0^{+}}B_{r}^{-1}(A(t))=B^{-1}(A(t))=u(t).
\]
\end{proof}

\begin{theorem}\label{Thm:2}
 Suppose that $b$ is non-decreasing, and $B(\infty )<A(\infty )$.
Then the solution of \eqref{Eq:EDM} explodes in finite time.
The time of explosion of $Y$ is
$t_e=A^{-1}\big( \overline{B}(\infty )\big) $.
\end{theorem}

\begin{proof}
The fact that $b$ is non-decreasing yields
\[
Y_{t}=x_0+\int_0^{t}a(s)b(Y_{s}+g(s))ds\geq
x_0+\int_0^{t}a(s)b(Y_{s})ds.
\]
Theorem \ref{Thm:Comparison} gives that $T_e\leq A^{-1}(B(\infty ))$
when we compare $Y$ with the solution of
$\widetilde{Y}_{t}=x_0+\int_0^{t}a(s)b(\widetilde{Y}_{s})ds$.
On the other hand,
$A(t_e)= \overline{B}(\infty )\leq B(\infty )<A(\infty )$,
which implies that $t_e$ is real. The expression of $t_e$ and
Corollary \ref{Coro:1} yield $A(t_e)=\overline{B}(Y_{T_e})=A(T_e)$.
We conclude that $T_e=t_e$.
\end{proof}

Intuitively, it is not generally true that the explosion of the
solutions of \eqref{Eq:EDM} in finite time is a sufficient
condition for the inequality $ B(\infty )<A(\infty )$ to be
satisfied. This assertion follows after noticing that $A$ and $B$
do not depend of the noise function; however, the time of
explosion does. We will establish our claim precisely through the
following counter-example.

\begin{example} \label{exa6} \rm
Let $x_0=1$, and let $a$, $b$ and $g$ be given by the expressions
$a(t)=e^{-t}$, $b(t)=\frac{1}{4}t^{3}$, and $g(t)=e^{t}$, for
every $t>0$.
Expanding the expression $(Y_{s}+e^{s})^{3}$ in \eqref{Eq:EIM},
we obtain $Y_{t}\geq 1+\frac{1}{4}\int_0^{t}Y_{s}^{2}ds$.
Then $Y_{t}\geq (1-\frac{1}{4}t)^{-1}$, which implies that $Y$
explodes in finite time. However, $B(\infty )=2>1=A(\infty )$.
\end{example}

The following result is a partial converse of Theorem \ref{Thm:2}.
We let $\widehat {g} (t) = \sup \{ g (s) : s \in [0 , t] \}$,
for every $t \geq 0$.

\begin{proposition}\label{Prop:1}
Suppose that $b$ is non-decreasing, and that
\[
\widehat{g}(t)<b(x_0)\int_{t}^{\infty }a(s)ds.
\]
If the solution $Y$ of \eqref{Eq:EIM} explodes in finite time, then
$B(\infty )<A(\infty )$.
\end{proposition}

\begin{proof}
By Corollary \ref{Coro:1},
$\overline{B}(\infty )=\overline{B}(Y_{T_e})=A(T_e)$.
Since $b$ is non-decreasing and $g(Y_{s}^{-1})\leq
\widehat{g}(T_e)$ for every $s\in [ x_0,\infty )$, we obtain
\[
\int_{x_0}^{\infty }\frac{ds}{b(s+\widehat{g}(T_e))}\leq A(T_e).
\]
Separating the integral in the definition of $B(\infty )$ as the sum
of the integrals over the intervals $[x_0,x_0+\widehat{g}(T_e)]$
and $[x_0+ \widehat{g}(T_e),\infty )$, using the facts that $b$
is positive and non-decreasing, and employing the last inequality
and the hypothesis, we obtain
\begin{align*}
B(\infty )
&\leq \int_{x_0}^{x_0+\widehat{g}(T_e)}\frac{ds}{b(x_0)}
+\int_{x_0}^{\infty }\frac{ds}{b(s+\widehat{g}(T_e))} \\
&\leq \frac{\widehat{g}(T_e)}{b(x_0)}+A(T_e)<A(\infty ).
\end{align*}
\end{proof}

\section{Approximation of the explosion time\label{S:Approx}}

It is important to notice that the time of explosion of $Y$,  as
given by the Theorem \ref{Thm:2}, presents the disadvantage of
depending on the solution $ Y$ itself. In this section, we will
derive some approximations to $T _e$ which do not present this
shortcoming. For the remainder of this manuscript and for the sake
of convenience, we let $T = A ^{- 1} (B (\infty))$. Throughout
this section, $b$ will be a non-decreasing function.

The Comparison criterion and Osgood's test yield that the time  of
explosion of the solution $Y$ of \eqref{Eq:EIM} satisfies
$T_e\leq T$. On the other hand,
\[
Y_{t}\leq x_0+\int_0^{t}a(s)b(Y_{s}+\widehat{g}(T))ds,
\]
and the Comparison criterion leads us to conclude that
\begin{equation}
A^{-1}\big( \beta (\widehat{g}(T))\big) \leq T_e\leq T.
 \label{Eq:EPA}
\end{equation}

In general, the function $b : [0 , \infty) \to \mathbb{R}$
is \emph{sub-multiplicative} if there exists a positive constant
$c$ such that $b (x y) \leq c b (x) b (y)$, for every $x , y \geq 0$.
Evidently, exponential and
power functions are sub-multiplicative.

Suppose that $b$ is a sub-multiplicative function, and let $c$ be the
positive number provided by the definition of sub-multiplicativity.
In the following, it will be convenient to define the function
$\widetilde{A} :[0,\infty )\to \mathbb{R}$ by
\[
\widetilde{A}(t)=c\int_0^{t}a(s)b\Big( \frac{1}{x_0}g(s)+1\Big) ds.
\]
This function is nonnegative, continuous and increasing and, thus, it is
invertible, and has a continuous and increasing inverse.

\begin{proposition}\label{Prop:2}
 Let $b$ be a sub-multiplicative function. Then $T _e \geq
\widetilde {A} ^{- 1} (B (\infty))$.
\end{proposition}

\begin{proof}
Since $Y_{s}\geq x_0$ for every $s\geq 0$, we obtain
\begin{align*}
g(s)+Y_{s}
&= Y_{s}g(s)\Big( \frac{1}{Y_{s}}+\frac{1}{g(s)}\Big) \\
&\leq Y_{s}g(s)\Big( \frac{1}{x_0}+\frac{1}{g(s)}\Big)
=Y_{s}\Big(\frac{1}{x_0}g(s)+1\Big) .
\end{align*}
Monotonicity and sub-multiplicativity of $b$, along
with \eqref {Eq:EIM},
yield
\[
Y_{t}\leq x_0+c\int_0^{t}a(s)b\Big( \frac{1}{x_0}g(s)+1\Big)
b(Y_{s})ds.
\]
The conclusion of the proposition follows now from the Comparison
criterion and Osgood's test.
\end{proof}

\section{An application\label{S:Applic}}

Throughout this section, we consider the stochastic differential
equation \eqref{Eq:Model} with noise function $|W_{t}|$,  where
$W$ is the Brownian motion. The noise function is taken in
absolute value in view of physical considerations on the dynamics
of cracks growth under fatigue loading. In fact, it has been
established experimentally that cracks in the subcritical stage
grow with a velocity that increases with the crack length
\cite{Paris}. The governing equation is called \emph{Paris'
equation}, and it is a power law (which is a sub-multiplicative
function) in which the exponent is determined empirically. As a
matter of fact, it has been established that Paris' law is valid
for a wide range of materials \cite{Allen1, Allen2, Mach, Wei}.

For the remainder of this work, we let $\Phi (x)$ represent the
probability that a random variable with standard normal
distribution assumes values in $ [-x , x]$, for every $x \geq 0$.

\begin{proposition} \label{prop9}
Let $0 \leq t < T$. Then
\begin{equation}  \label{Eq:PRA1}
P (T _e \leq t) \leq 1 - \Phi
\Big( \frac {\beta ^{- 1} (A (t))} {\sqrt {T}}\Big).
\end{equation}
\end{proposition}

\begin{proof}
We use here the first inequality of \eqref{Eq:EPA}. Notice that
\[
P( T_e\leq t) \leq P\big( A^{-1}(\beta (|\widehat{W}
_{T}|))\leq t\big)
=P\big( |\widehat{W}_{T}|\geq \beta ^{-1}(A(t))\big).
\]
Then \cite[Section 2.8, Eq. (8.4)]{Karatzas} completes the proof.
\end{proof}

For every nonnegative, real number $r$, we let
$T _r = \inf \{ t > 0 : | W _t | = r \}$. Evidently,
$| W _s | \leq r$, for every $s \in [0 , T _r]$.

\begin{proposition}\label{Prop:4}
Let $0 \leq t \leq T$. For every $r \geq 0$,
\begin{equation}  \label{Eq:PRA2}
P \left( T _e \leq t | T _r < T \right)
\leq \frac {1 - \Phi \big(r /\sqrt {A ^{- 1}
(B (\widetilde {B} ^{- 1} _r (A (t))))}\big)}
{1 - \Phi \big(r /\sqrt {T}\big)}.
\end{equation}
\end{proposition}

\begin{proof}
Notice that $|\widehat{W}_{T}|\geq r$ whenever $T_{r}<T$.
Moreover, Osgood's test and the Comparison criterion imply
that $Y_{t}\geq B^{-1}(A(t))$, for
every $t\geq 0$. Using \eqref{Eq:EPA}, we obtain
$A(T_e)\geq \widetilde{B} _{|\widehat{W}_{T}|}(\infty )
\geq \widetilde{B}_{r}(\infty )
\geq \widetilde{B}_{r}(B^{-1}(A(T_{r})))$.
Therefore,
\begin{align*}
P( T_e\leq t|T_{r}<T)
&\leq  \frac{P\big( \widetilde{B}
_{r}(B^{-1}(A(T_{r})))\leq A(t)\big) }{P(T_{r}<T)} \\
&= \frac{P\big( T_{r}<A^{-1}(B(\widetilde{B}_{r}^{-1}(A(t))))\big) }{
P(T_{r}<T)}.
\end{align*}
The conclusion follows now from \cite{Karatzas} as in Proposition
\ref{prop9}.
\end{proof}

On physical grounds, the function $Y$ may represent the  temporal
behavior of the transversal length of a crack failure on some
material \cite{Paris}. In this context, the parameter $x _0$
represents the initial, transversal length of the crack, and $L$
is the transversal length of the material. For practical purposes,
one may think of the wing of an airplane which has a fixed
transversal length, on which there is a crack with known initial
length. In such case, one investigates the dynamics of the length
of the crack with respect to time, in order to conduct preventive
maintenance on the wing and avoid possible accidents
\cite{Augustin}.

\begin{proposition} \label{p11}
If $L>x_0$, then
\[
P( Y_{L}^{-1}\leq t) \leq 1-\Phi \Big( \frac{\widetilde{B}
_{A(t)}^{-1}(L)}{\sqrt{T}}\Big) .
\]
\end{proposition}

\begin{proof}
Let $\widetilde{Y}$ the solution of $\widetilde{Y}_{t}=x_0+
\int_0^{t}a(s)b(\widetilde{Y}_{s}+|\widehat{W}_{T}|)ds$, for
every $0\leq t<T$. By Osgood's test and the Comparison criterion,
$\widetilde{B}_{|\widehat{W}_{T}|}^{-1}(A(t))=\widetilde{Y}_{t}
\geq Y_{t}$. Once again, the
conclusion is reached using \cite{Karatzas} in the right-most
end of the chain of identities and inequalities
\begin{align*}
P( Y_{L}^{-1}\leq t)
&\leq P\big( \widetilde{Y}_{L}^{-1}\leq t\big) \\
&= P\big( \widetilde{B}_{|\widehat{W}_{T}|}(L)\leq A(t)\big) \\
&= 1-P\big( |\widehat{W}_{T}|\leq \widetilde{B}_{A(t)}^{-1}(L)\big) .
\end{align*}
\end{proof}

\begin{example} \label{exa12} \rm
Let $x_0$, $a_0$ and $\alpha $ be positive numbers, and let
$a(t)=a_0$ and $b(t)=t^{1+\alpha }$, for every $t\geq 0$. Observe
that $A(t)=a_0t$ and, for every $r\in [ 0,x_0]$ and every $x\geq x_0$,
\[
B_{r}(x)=\frac{1}{\alpha }\big[ \frac{1}{(x_0-r)^{\alpha }}-\frac{1}{
x^{\alpha }}\big] ,
\]
so that $T=(\alpha a_0x_0^{\alpha })^{-1}$. By \eqref{Eq:PRA1},
\begin{equation}
P\left( T_e\leq t\right)
\leq 1-\Phi \Big( \frac{(\alpha
a_0t)^{-1/\alpha }-x_0}{\sqrt{T}}\Big) ,  \label{Eq:AFE}
\end{equation}
for every $0\leq t<T$. In order to estimate the value of $t$ for which
$T_e\leq t$ with a probability of at most $0.05$,
Equation \eqref{Eq:AFE} yields
\[
1-\Phi \Big( \frac{(\alpha a_0t)^{-1/\alpha }-x_0}{\sqrt{T}}\Big)
\leq 0.05
\]
whence it follows that $t=\frac{1}{\alpha a_0}[x_0+\sqrt{T}\Phi
^{-1}(0.95)]^{-\alpha }$.
Proposition \ref{Prop:2} and monotonicity on the
integrand imply that
\begin{align*}
P( T_e\leq t)
& \leq {P\Big(B(\infty )\leq \widetilde{A}(t)\Big) } \\
& \leq   {P\Big( \frac{1}{\alpha x_0^{\alpha }}\leq
\int_0^{t}a_0\big( \frac{1}{x_0}|\widehat{W}_{t}|+1\big) ^{1+\alpha
}ds\Big) } \\
& \leq  {P\Big( \frac{1}{\alpha x_0^{\alpha }}\leq a_0t\big(
\frac{1}{x_0}|\widehat{W}_{t}|+1\big) ^{1+\alpha }\Big) }
\\
& = {1-\Phi \Big( \frac{x_0}{\sqrt{t}}\big( (\alpha
a_0x_0^{\alpha }t)^{-1/(1+\alpha )}-1\big) \Big) }.
\end{align*}
This last estimate of $P(T_e\leq t)$ is better than that given
 by \eqref{Eq:AFE}, in view of the fact that
$\alpha a_0x_0^{\alpha }t>1$.
\end{example}

\subsection*{Acknowledgments}
M. J. Ceballos-Lira wishes to acknowledge the financial support of
the Mexican Council for Science and Technology (CONACYT) to pursue
postgraduate studies in Universidad Ju\'arez Aut\'onoma de Tabasco
(UJAT); he also wishes to thank UJAT and the Universidad
Aut\'onoma de Aguascalientes (UAA) for additional, partial,
financial support. J. Villa acknowledges the partial support of
CONACYT grant 118294, and grant PIM08-2 at UAA.

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\end{document}