## DSE

Hello,
I do have a big problem, can anyone help me?

$\forall x \in \mathbb{R}$ we consider the SDE : $X_t^x=x+\int_{0}^{t}{\sigma(X_s^x) dB_s}$

for every $[y,z]$, the stoping time sequence : $\tau^x_{(y,z)}=inf(t \geqslant 0 : X_t^x \notin ]y,z[)$

First we shox that for every $\phi \in \mathcal{C}^{2}(\mathbb{R})$ with compact support, :

$\mathbb{E}(\phi(X_{\min(t,\tau^x)}^x))=\phi(x)+\fr ac{1}{2}\mathbb{E}(\int_{0}^{\min(t,\tau^x)}{a.\ph i^{\prime \prime}(X_s^x) ds})$
(with $a(x)=\sigma^2(x)$) thank to the Itô formula.

then my problem is to show that $\tau^x<\infty$a.s. by choosing a "good" $\phi$ function.