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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 140, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/140\hfil Explosion time]
{Explosion time in stochastic differential equations with
small diffusion}

\author[P. Groisman, J. D. Rossi\hfil EJDE-2007/140\hfilneg]
{Pablo Groisman, Julio D. Rossi}  % in alphabetical order

\address{Pablo Groisman \newline
Instituto de C\'alculo, FCEyN, Universidad de Buenos Aires,
Pabell\'on II, Ciudad Universitaria (1428),
Buenos Aires, Argentina}
\email{pgroisma@dm.uba.ar}
\urladdr{http://mate.dm.uba.ar/$\sim$pgroisma}


\address{Julio D. Rossi \newline
Instituto de Matem\'aticas y F\'{\i}sica Fundamental,
 Consejo Superior de Investigaciones
Cient\'{\i}ficas, Serrano 123, Madrid, Spain \hfill\break
Departamento de Matem\'atica, FCEyN UBA (1428),
 Buenos Aires, Argentina}
\email{jrossi@dm.uba.ar}
\urladdr{http://mate.dm.uba.ar/$\sim$jrossi}

\thanks{Submitted June 21, 2007. Published October 19, 2007.}
\subjclass[2000]{60H10, 60G17, 34F05}
\keywords{Explosion; stochastic differential equations}

\begin{abstract}
 We consider solutions of a one dimensional stochastic differential
 equations that explode in finite time. We prove that, under
 suitable hypotheses, the explosion time converges almost surely to
 the one of the ODE governed by the drift term when the diffusion
 coefficient approaches zero.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Explosions in one dimensional ODEs is a very well known phenomena.
Let $u(t)$ be the solution of
\begin{equation}\label{ode.b}
\dot u = b(u), \quad u(0)=x_0.
\end{equation}
If $b(\cdot)>0$, there exists a
finite time $T$ such that $\lim_{t \nearrow T} u(t) = +\infty$ if
and only if $\int^\infty 1/ b < +\infty$. In this case we have an
explicit formula for the explosion time $T$ in terms of $b$ and
$x_0$,
\begin{equation}\label{T}
T = \int_{x_0}^\infty \frac 1 {b(s)} \, ds.
\end{equation}
On the other hand, let us consider the stochastic differential
equation
\begin{equation}\label{SDE.ito}
dX = b(X)\, dt + \sigma(X) \, dW, \quad X(0)=x_0 >0,
\end{equation}
where $b$ and $\sigma$ are smooth positive functions and $W$ is a (one
dimensional) Wiener process defined on a given probability space
$(\Omega, \mathbb{P})$.

As happens with \eqref{ode.b}, solutions of \eqref{SDE.ito} may
explode in finite time, that is, trajectories may diverge to
infinity as $t$ goes to some finite time $S$ that in general
depends on the particular sample path.

This phenomena has been considered, for example, in fatigue
cracking (fatigue failures in solid materials) with $b$ and $\sigma$
of power type, see \cite{S}. In this case the explosion time
corresponds to the time of ultimate damage or fatigue failure in
the material.


The \emph{Feller Test for explosions} (see \cite{KS, MK}) gives a
precise description in terms of $b$, $\sigma$  and $x_0$ of
whether explosions in finite time occur with probability zero,
positive or one. For example, if $b$ and $\sigma$ behave like
powers at infinity; i.e., $ b(s) \sim s^p$, $\sigma(s) \sim s^q$ as
$s\to \infty$, applying the Feller test one obtains that solutions
to \eqref{1.1} explode with probability one if and only if
$p> 2q- 1$ and $p>1$.


There is no simple formula for the explosion time $S$ as
\eqref{T} (although there exists some expressions for a version of
$S$ that involve the scale function which can be found in
\cite{KS}). Hence, to estimate $S$ is a nontrivial task.
In order to get information about the stochastic explosion time
one can use adaptive numerical approximations like the ones
described in \cite{DFBGRS} where the authors provide a numerical
method that can be used to compute a convergent approximation of
$S$.

In this article we find, by theoretical arguments, estimates on
the explosion time $S$ when the diffusion $\sigma$ is small. That is,
we look at \eqref{SDE.ito} as a stochastic perturbation of the ODE
\eqref{ode.b}. We prove that, under adequate hypotheses on $b$ and $\sigma$,
the stochastic explosion time, $S=S(\sigma)$, converges to the
deterministic one, $T$, almost surely when $\sigma$ goes to zero. This
means that the stochastic explosion times converge to a constant,
$T$ given by
\eqref{T}, that can be explicitly computed.


In the statement of the theorem we use the Stratonovich integral
since the proofs are simpler. This is not a restriction thanks to
the well known conversion formula (see below). We consider a
family of SDE
\begin{equation}\label{SDE.strato}
dX = b(X)\, dt + \sigma(X,\varepsilon) \circ dW, \quad X(0)=x_0 >0,
\end{equation}
where $\varepsilon>0$ is a parameter and $\sigma(\cdot,\varepsilon) \to 0$ as
$\varepsilon \to 0$. We introduce a function
$H \colon \mathbb{R} \times \mathbb{R}_+ \times \mathbb{R}_+
\to \mathbb{R}$ ($\mathbb{R}_+=[0,+\infty))$ defined in this way: Let
$\phi=\phi_\varepsilon(t,x)$ the flux associated to the ODE
\begin{equation}
\label{ode.sigma.epsilon}
\dot y = \sigma(y,\varepsilon), \quad y(0)=x.
\end{equation}
We assume that $\sigma(\cdot,\varepsilon)$ is globally Lipschitz and smooth
and therefore $\phi_\varepsilon$ is globally defined. Then we define
$$
H(s,x,\varepsilon) = \frac{b(\phi_\varepsilon(s,x))
\sigma(x,\varepsilon)}{\sigma(\phi_\varepsilon(s,x),\varepsilon)}.
$$


\begin{theorem}\label{teo1} Assume
\begin{enumerate}
\item $b>0$ in $\mathbb{R}_+$ and $\sigma (\cdot , \varepsilon )>0$ has continuous bounded derivatives in
$\mathbb{R}_+$;

\item   Given $s \in \mathbb{R}$, there exists $g_s \in
L^1(\mathbb{R}_+)$ such that for every $x \in \mathbb{R}_+$,
\begin{equation} \label{mayorada}
\frac 1 {H(s,x,\varepsilon)} \le g_s(x);
\end{equation}


\item $ H(s, x, \varepsilon) \ge H(t, x, \varepsilon)$  if $s \ge t$,

\item $\lim_{\varepsilon \to 0} H(s, x, \varepsilon) = b(x)$;
\end{enumerate}
then for almost every $\omega$ the (strong) solution of
\eqref{SDE.strato} explodes in finite time $S_\varepsilon(\omega)$ for
every $\varepsilon >0$ and
\begin{equation}\label{convergencia}
\lim_{\varepsilon \to 0} S_\varepsilon(\omega) = T.
\end{equation}
If in addition $H$ satisfies
\begin{enumerate}
\item[(5)] For every $s \in \mathbb{R}$, there exists $f_s \in
L^1(\mathbb{R}_+)$ such that $\frac{\partial}{\partial \varepsilon} \frac 1
{H(s,x,\varepsilon)} \le f_s(x)$ for every $x \in \mathbb{R}_+$ and $0<\varepsilon
<\varepsilon_0$,

\end{enumerate}
then $S_\varepsilon(\omega)$ is Lipschitz continuous at $\varepsilon=0$ almost
surely, that is, there exist a random variable $C=C(\omega)$ such
that with total probability
$$
|T-S_\varepsilon(\omega)| \le C \varepsilon.
$$
\end{theorem}

\begin{remark} \label{rm1.1} \rm
 If a SDE is given in It\^o form,
we can apply the conversion formula: $X(t)$ solves $dX = f(X)dt + g(X) dW$
if and only if it solves \eqref{SDE.strato} with $b=f - \frac12 \sigma'\sigma$, $\sigma=g$.
 In this case we obtain that also $b$ depends on $\varepsilon$ but similar
 results can be obtained (see the second part of Example \ref{exa2}).
\end{remark}

\begin{remark} \label{rm1.2} \rm
If $b(x)/\sigma (x,\varepsilon)$ is increasing in $x$ then the monotonicity of
$H(s,x,\varepsilon)$ in $s$, hypothesis
(3), holds.
\end{remark}

\begin{remark} \label{rm1.3} \rm
If $H(s,x, \varepsilon)$ is increasing (or decreasing) in $\varepsilon$
then we can get rid of hypothesis (2), using the Monotone
Convergence Theorem instead of the Dominated Convergence Theorem
in the proof.
\end{remark}

\section{Some simple examples}

In this section we consider some simple examples to illustrate the
main ideas used in the proof of Theorem \ref{teo1} and the
principal features of the problem. We do not invoke Theorem
\ref{teo1} to deal with these examples, we prove the results ``by
hand". We are going to make use of Theorem \ref{teo1} in the
examples of  the last section.

The main idea is to change variables in order to transform the SDE
into a random differential equation. Then we obtain bounds for the
explosion time by using sub and supersolutions given by ODEs.


\begin{example}[Aadditive noise] \label{exa1} \rm
 Let $u(t)$ be the solution of
\eqref{ode.b} with $b$ increasing and $\int^\infty 1/b <+\infty$.
Let $X$ be a solution of the It\^o SDE
$$
dX = b(X) dt + \varepsilon dW, \quad X(0)=x_0.
$$
Note that in this particular case It\^o and Stratonovich
interpretations are identical.
Let $Z= X - \varepsilon W$, then $Z$ solves
$$
dZ = dX - \varepsilon dW = b(Z + \varepsilon W) dt, \quad  Z(0)=x_0.
$$
This gives a non-autonomous ODE for each $\omega$ such that
$W(\cdot,\omega)$ is continuous,
\begin{equation}\label{Z.omega.2.1}
\dot Z_\omega(t) = b(Z_\omega(t) + \varepsilon W(t,\omega)), \quad
Z_\omega(0) = x_0.
\end{equation}
In this equation $\omega$ is regarded as a parameter.

Given $M>0$, we consider $\overline z$ and $\underline z$ the
solutions of
$$
\dot{\overline z}(t) = b(\overline z(t) + \varepsilon M), \quad \overline
z(0) = x_0
$$
and
$$
\dot{\underline z}(t) = b(\underline z(t) - \varepsilon M), \quad
\underline z(0) = x_0.
$$
These solutions explode in finite time given by
$$
\overline T_\varepsilon = \int_{x_0}^\infty \frac 1 {b(s + \varepsilon M)} \, ds,
\quad \underline T_\varepsilon = \int_{x_0}^\infty \frac 1 {b(s - \varepsilon M)}
\, ds,
$$
respectively. Since $b$ is increasing, by the Monotone Convergence
Theorem we get
\begin{equation}\label{Tsub.Tsuper}
\lim_{\varepsilon \to 0} \overline T_\varepsilon = \lim_{\varepsilon \to 0} \underline
T_\varepsilon = T.
\end{equation}
Let
$$
A_M = \big\{ \omega \colon W(\cdot, \omega) \mbox{ is continuous
and }\max_{0\le t \le T+1} |W(\cdot,\omega)|
\le M \big\}.
$$
For $\omega \in A_M$, $\overline z$ and $\underline z$ are super
and subsolutions of \eqref{Z.omega.2.1} for $0<t<T+1$. Using
\eqref{Tsub.Tsuper}, a comparison argument gives
$$
\underline z(t) \le Z_\omega(t) \le \overline z(t),
$$
as long as all of them are defined. Hence, for $\omega \in A_M$,
$$
\overline T_\varepsilon \le S_\varepsilon(\omega) \le \underline T_\varepsilon.
$$
Therefore, by \eqref{Tsub.Tsuper},
$$
\lim_{\varepsilon \to 0} S_\varepsilon(\omega) = T.
$$
As
$$\mathbb{P} \big (\bigcup_{M=1}^\infty A_M \big ) = 1 $$ we get the desired
result.

\begin{remark} \label{rmk2.1} \rm
In this example the function $H$ involved in Theorem
\ref{teo1} is given by
$$
H(s,x,\varepsilon) = b(x+\varepsilon s),
$$
and verifies the hypotheses stated there.
\end{remark}

Observe also that in the ODE \eqref{ode.b}, the function $b$ does
not need to be increasing in order to have explosions. In this
example, the monotonicity of $b$ is only used to take limits in
\eqref{Tsub.Tsuper}, but we can get rid of this hypothesis if we can
ensure that those limits hold.
\end{example}

\begin{example}[Multiplicative noise] \label{exa2} \rm
 Let $u(t)$ be as in Example
1. Let $X$ be the solution of the Stratonovich SDE
$$
dX = b(X) dt + \varepsilon X \circ dW, \quad X(0)=x_0.
$$
As in the preceding example, we want to get an ODE for each
$\omega$. To do that, let $Z= X e^{-\varepsilon W}$. Hence we get that $Z$
solves
$$
dZ = \left( e^{-\varepsilon W} b(Z e^{\varepsilon W}) \right) dt, \quad Z(0)=x_0.
$$
As before, this gives a non-autonomous ODE for each $\omega$ such
that $W(\cdot,\omega)$ is continuous,
\begin{equation}\label{Z.omega.2.2}
\dot Z_\omega(t) = e^{-\varepsilon W(t,\omega)} b(Z_\omega(t) e^{\varepsilon
W(t,\omega)})  , \quad Z_\omega(0) = x_0.
\end{equation}
Given $M>0$, we consider $\overline z$ and $\underline z$ the
solutions of
$$
\dot{\overline z}(t) = e^{\varepsilon M} b(\overline z(t)e^{\varepsilon M})  ,
\quad \overline z(0) = x_0
$$
and
$$
\dot{\underline z}(t) = e^{-\varepsilon M} b(\underline z(t) e^{-\varepsilon M}) ,
\quad \underline z(0) = x_0.
$$
These solutions explode in finite time given by
$$
\overline T_\varepsilon
=
\int_{x_0}^\infty \frac 1 {e^{\varepsilon M} b( s e^{\varepsilon M}) } \, ds,
\quad \underline T_\varepsilon = \int_{x_0}^\infty \frac 1 {e^{-\varepsilon M}
b(s e^{-\varepsilon M}) } \, ds,
$$
respectively. We have
\begin{equation}\label{Tsub.Tsuper2}
\lim_{\varepsilon \to 0} \overline T_\varepsilon = \lim_{\varepsilon \to 0} \underline
T_\varepsilon = T.
\end{equation}
Let $A_M$ as before. Since $b$ is increasing, for $\omega \in
A_M$, $\overline z$ and $\underline z$ are super and subsolutions
of \eqref{Z.omega.2.2} for $0<t<T+1$ and hence, using
\eqref{Tsub.Tsuper2}, we can compare their explosion times
$$
\overline T_\varepsilon \le S_\varepsilon(\omega) \le \underline T_\varepsilon.
$$
Therefore
$$
\lim_{\varepsilon \to 0} S_\varepsilon(\omega) = T.
$$
and we get the desired result. In this case $H(s,x,\varepsilon) = e^{-\varepsilon s}
b(x e ^{\varepsilon s})$.


Now, let us consider the same equation but in It\^o sense.
Let
$X$ be the solution of the It\^o SDE
$$
dX = b(X) dt + \varepsilon X dW, \quad X(0)=x_0.
$$
As before, we want to get an ODE for each $\omega$. To do that,
let $Z= X e^{-\varepsilon W}$. Using It\^o's rule we get
$$
dZ = \big( e^{-\varepsilon W} b(Z e^{\varepsilon W}) -\frac12 \varepsilon^2 Z \big) dt,
\quad Z(0)=x_0.
$$
Again this gives a non-autonomous ODE for each $\omega$ such that
$W(\cdot,\omega)$ is continuous,
\begin{equation}\label{Z.omega.2.3}
\dot Z_\omega(t) = e^{-\varepsilon W(t,\omega)} b(Z_\omega(t) e^{\varepsilon
W(t,\omega)}) -\frac12 \varepsilon^2 Z_\omega (t) , \quad Z_\omega(0) =
x_0.
\end{equation}
Given $M>0$, we consider $\overline z$ and $\underline z$ the
solutions of
$$
\dot{\overline z}(t) = e^{\varepsilon M} b(\overline z(t)e^{\varepsilon M}) -\frac12 \varepsilon^2 \overline{z} (t) ,
\quad \overline z(0) = x_0
$$
and
$$
\dot{\underline z}(t) = e^{-\varepsilon M} b(\underline z(t) e^{-\varepsilon M})  -\frac12 \varepsilon^2 \underline{z} (t),
\quad \underline z(0) = x_0.
$$
These solutions explode in finite time given by
$$
\overline T_\varepsilon =
\int_{x_0}^\infty \frac 1 {e^{\varepsilon M} b( s e^{\varepsilon M}) -\frac12 \varepsilon^2 s} \, ds,
\quad \underline T_\varepsilon = \int_{x_0}^\infty \frac 1 {e^{-\varepsilon M}
b(s e^{-\varepsilon M}) -\frac12 \varepsilon^2 s} \, ds,
$$
respectively. Since $1/b$ is integrable these times are finite and we can apply dominated convergence we obtain
\begin{equation}\label{Tsub.Tsuper2.2.1}
\lim_{\varepsilon \to 0} \overline T_\varepsilon = \lim_{\varepsilon \to 0} \underline
T_\varepsilon = T.
\end{equation}
{}From this point the limit
$$
\lim_{\varepsilon \to 0} S_\varepsilon(\omega) = T
$$
follows exactly as before.

In this example the function $H$ is
$$
H(s,x,\varepsilon) =e^{-\varepsilon s} b(x e^{\varepsilon s}) -\frac12 \varepsilon^2 x.
$$
Observe that since $b$ is superlinear $H$ is increasing in time.
However this hypothesis is not required in this case. The result
can also be obtained since we can bound $H$ from above and from
below by functions that converge to $b$ as $\varepsilon \to 0$.
\end{example}

\section{Proof of the main result}

\subsection*{Pathwise solutions of the SDE}
We want to apply the same ideas used in the previous examples,
that is, to transform the SDE in a non-autonomous ODE where
$\omega$ plays the role of a parameter. This is easier when the
equation is understood in Stratonovich sense.

The study of pathwise solutions to stochastic differential
equations via an appropriate reduction to an ODE was first done in
\cite{D,Su}. We refer to those works and to \cite{KS} for details.

Consider a solution of the Stratonovich SDE
\begin{equation}\label{1.1}
dX = b(X) dt + \sigma(X) \circ dW.
\end{equation}
This solution may explode in finite time or may be globally
defined.

Let $y$ be a solution of the ODE
\begin{equation}
\label{ode.sigma} \dot y = \sigma(y), \quad y(0)=x,
\end{equation}
and let $\phi(t,x)$ the flux associated to \eqref{ode.sigma} which
is globally defined and has continuous derivatives, since $\sigma$ is
smooth and globally Lipschitz.

Consider $Z_\omega=Z_\omega(t)$ the solution of the random
differential equation
\begin{equation}\label{random}
\begin{gathered}
  d Z_\omega(t) =
\frac{b(\phi(W(t,\omega),Z_\omega(t)))}{\phi_x(W(t,\omega),Z_\omega(t))} \,dt, \\
Z_\omega(0)=x_0.
\end{gathered}
\end{equation}
Then $X(t,\omega)=\phi(W(t,\omega),Z_\omega(t))$ is a strong
solution of
\eqref{1.1} up to a possible explosion time $S_\varepsilon$. In fact,
since \eqref{1.1} is interpreted in Stratonovich sense, we have
\begin{gather*}
dX = \phi_t(W,Z_\omega) \, dW + \phi_x(W,Z_\omega) \, d Z_\omega =
\sigma(X) \, dW + b(X) \, dt, \\
 X(0)=x_0.
\end{gather*}
Note that the explosion time $S_\varepsilon(\omega)$ is the maximal
existence time of \eqref{random} for each $\omega$. We are going
to use this fact to prove Theorem \ref{teo1}.


\begin{proof}[Proof of Theorem \ref{teo1}]
 First of all observe that
assumptions (1) and (2) ensure on the one hand that solutions to
\eqref{ode.b},\eqref{SDE.strato} are positive and on the other
hand that solutions to \eqref{ode.b} explodes in finite time $T$
given by \eqref{T}. Applying the Feller Test for explosions one
can see that these hypotheses also ensure that
\eqref{SDE.strato} explodes in finite time with probability one.
Nevertheless we are going to show this fact in the course of the
proof.

For each $\omega$ such that $W(\cdot,\omega)$ is continuous,
consider the ODE
\begin{equation}\label{Z.omega.teo}
\dot Z_\omega(t) =
\frac{b(\phi_\varepsilon(W(t,\omega),Z_\omega(t)))}{(\phi_\varepsilon)_x(W(t,\omega),Z_\omega(t))},
\quad Z_\omega(0)=x_0.
\end{equation}
Here $\phi_\varepsilon$ is the flux associated to the ODE
\eqref{ode.sigma.epsilon}. The equation \eqref{Z.omega.teo} can be
written in terms of $H$ as
\begin{equation}\label{Z.H}
\dot Z_\omega(t) = H(W(t,\omega),Z_\omega(t),\varepsilon), \quad
Z_\omega(0)=x_0.
\end{equation}
In fact, integrating \eqref{ode.sigma.epsilon} we get
$$
\int_{x}^{\phi_\varepsilon(t,x)} \frac{ d\tau}{\sigma(\tau, \varepsilon)} = t.
$$
Differentiating with respect to $x$ we obtain
$$
\frac{(\phi_\varepsilon)_x(t,x)}{\sigma(\phi_\varepsilon(t,x),\varepsilon)} - \frac 1
{\sigma(x,\varepsilon)} = 0,
$$
hence
$$
(\phi_\varepsilon)_x(t,x) =    \frac {\sigma(\phi_\varepsilon(t,x),\varepsilon)} {\sigma(x,\varepsilon)}
$$
and so
$$
H(s,x,\varepsilon) = \frac{b(\phi_\varepsilon(s,x))}{(\phi_\varepsilon)_{x} (s,x)}.
$$
Given $M>0$, we consider $\overline z$ and $\underline z$ the
solutions of
$$
\dot{\overline z}(t) = H(M,\overline z(t),\varepsilon), \quad \overline
z(0) = x_0
$$
and
$$
\dot{\underline z}(t) = H(-M,\underline z(t),\varepsilon), \quad
\underline z(0) = x_0.
$$
These solutions explode in finite time given by
$$
\overline T_\varepsilon=
\int_{x_0}^\infty \frac 1 {H(M,x,\varepsilon)} \, dx,
\quad \underline T_\varepsilon = \int_{x_0}^\infty \frac 1 {H(-M,x,\varepsilon)}
\, dx,
$$
respectively. By assumption \eqref{mayorada} we can apply the
Dominated Convergence Theorem to get
\begin{equation}\label{Tsub.Tsuper.teo}
\lim_{\varepsilon \to 0} \overline T_\varepsilon = \lim_{\varepsilon \to 0} \underline
T_\varepsilon = T.
\end{equation}

Let $A_M$ be as in the examples. Since $H(s,x,\varepsilon)$ is increasing
in the $s$ variable, for any $\omega \in A_M$, $\overline z$ and
$\underline z$ are super and subsolutions of \eqref{Z.H} for
$0<t<T+1$. Using this fact and \eqref{Tsub.Tsuper.teo}, their
explosion times can be compared. Since
$X(t)=\phi_\varepsilon(W(t),Z_\omega(t))$ and $\phi_\varepsilon$ is globally
defined, the explosion times of $X$ and $Z_\omega$ coincide a.s.
Then we obtain
$$
\overline T_\varepsilon \le S_\varepsilon(\omega) \le \underline T_\varepsilon.
$$
Therefore
$$
\lim_{\varepsilon \to 0} S_\varepsilon(\omega) = T.
$$
As
$$\mathbb{P} \big (\bigcup_{M=1}^\infty A_M \big ) = 1,$$
we have proved \eqref{convergencia}. It remains to show the Lipschitz
continuity. To do this observe that the Taylor expansion of
$1/H(\pm M,x,\varepsilon)$ at $\varepsilon=0$ gives for some $\eta_\varepsilon$ with $0<\eta_\varepsilon<\varepsilon$,
\begin{align*}
|S_\varepsilon(\omega) - T|
& \le   \big|\int_{x_0}^\infty \frac 1
{H(\pm M,x,\varepsilon)} \, dx - T \big|\\
& =   \big|\int_{x_0}^\infty \frac 1 {b(x)} \, dx + \int_{x_0}^\infty \varepsilon
\frac{\partial}{\partial \varepsilon} \frac 1
{H(\pm M,x,\eta_\varepsilon)} \, dx - T \big|\\
& =   \big| \int_{x_0}^\infty \varepsilon
\frac{\partial}{\partial \varepsilon} \frac 1
{H(\pm M,x,\eta_\varepsilon)} \, dx \big| \\
& \le   \varepsilon \big| \int_{x_0}^\infty f_M (x) \, dx \big| \\
& \le  C \varepsilon.
\end{align*}
This completes the proof.
\end{proof}

\section{More Examples}

In this section we present two additional examples where the
result can be applied.


\begin{example}[Unbounded diffusion] \label{exa3} \rm
Let $u(t)$ be the solution of
\eqref{ode.b} and consider the SDE
$$
dX = b(X) dt + \varepsilon \sigma (X) \circ dW, \quad X(0)=x_0,
$$
with
$$
b (x) \sim x^p, \quad \sigma (x) \sim x^q,  \quad 0<q<1<p,
$$
for large $x$ and bounded below away from zero.
In this case we have
$$
\phi_\varepsilon(t,x) \sim \left(x^{1-q} + (1-q)\varepsilon t \right
)^{\frac{1}{1-q}},
$$
for $x$ large and $t>0$. Hence, the behavior of  $H(s,x,\varepsilon)$ at
infinity is given by
$$
H(s,x,\varepsilon) =  \frac{b(\phi_\varepsilon(s,x))
\sigma(x,\varepsilon)}{\sigma(\phi_\varepsilon(s,x),\varepsilon)} \sim \left(x^{1-q} + (1-q)\varepsilon
s\right )^{\frac{p-q}{1-q}}x^q \sim C x^p.
$$
{}From these expressions it is easy to check hypotheses (1), (2)
and (4). If we assume (3), then we can apply our theorem to get
$S_\varepsilon \to T$ almost surely. Note that (3) holds if we take, for example,
$b(x)=(1+|x|)^p$, $\sigma(x)=\varepsilon(1+|x|)^q$. In fact, for $x\ge0$ we have
$$
H(s,x,\varepsilon)= \left ( \varepsilon s(1-q) + (1+x)^{1-q}\right
)^\frac{p-q}{1-q}(1+x^q).
$$
In this case ,(5) also
holds and so $S_\varepsilon$ is Lipschitz at $\varepsilon=0$ almost surely.
\end{example}


\begin{example}[Bounded diffusion] \label{exa4} \rm
In this example we consider
$$ dX = b(X) dt +
\varepsilon \sigma (X) \circ dW, \quad X(0)=x_0,
$$
with a bounded $\sigma$, $ 0<c_1 \le \sigma  \le C_2$ and $b$ such that
$\int^\infty 1/ b < +\infty$. We have
$$
\frac{1}{H(s,x,\varepsilon)} \le g_s (x) := \frac{C}{b(x)}.
$$
If we assume that (3) holds (the rest of the hypotheses can be
easily checked) we obtain again that $S_\varepsilon \to T$ almost surely.
\end{example}


\begin{example} \label{exa5} \rm
 In this example we consider
$$
dX = e^{a X} dt +
\varepsilon e^{b X} \circ dW, \quad X(0)=x_0,
$$
with $a>b>0$.
In this case we have that the solution of
$$
\dot y = \sigma(y,\varepsilon), \quad y(0)=x
$$
is given by
$$
y(s) = \phi_\varepsilon (s,x) = \frac{ \ln ( -b \varepsilon s + e^{-bx} )}{-b}.
$$
Therefore, we obtain
$$
H (s,x,\varepsilon) = e^{ax} |1- b s \varepsilon e^{bx}|^{1-\frac{a}{b}}
$$
and we can conclude as before that $S_\varepsilon
\to T$ almost surely.
\end{example}

\subsection*{Acknowledgments}
The authors want to thank J. A. Langa for several
interesting discussions.
This research was supported by grant X066 from the Universidad
de Buenos Aires,  by grant  03-05009 from ANPCyT PICT,
by Fundacion Antorchas, and by CONICET (Argentina).

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\end{document}