Let $\sigma \in C(\mathbb R)$, and $X$ be a solution of
\begin{equation}\label{eq:1}
 X\_{t} = x + t + \int\_{0}^{t} \sigma(X\_{s}) dB\_{s} 
\end{equation}
where $B$ is 1-d Brownian motion under filtered probability space $(\Omega, \mathcal F, \mathbb P, \{\mathcal F\_{t}\}\_{t\ge 0})$.
We denote by $\mathbb P^{x}$ the probability on $C([0, \infty), \mathbb R)$ induced by the process $X$ starting from $x$.
Consider a stopping time $\tau = \inf\{t>0: X(t) <0\}$. 


[Q.] What is the sufficient condition on $\sigma$ to have the continuity of 
$u(x) = \mathbb E^{x}[e^{-\tau}]$?


[A.] $\sigma \in C(0,1)$ and $\sigma(0) \neq 0$.


[Sketch of Proof.] If $\sigma \in C^{0,1}$, then the following two conditions are standard:


(C1) The above SDE has unique strong solution;


(C2) $\mathbb P^{x\_{n}} \Rightarrow \mathbb P^{x}$ whenever $x\_{n} \to x$;


Moreover, according to the [Discussion 2] of the post ([regularity of zero point](https://mathoverflow.net/q/221196/5656)), we also have


(C3) $\mathbb P^{0} (\tau = 0) = 1$.


In fact, (C3) implies that the mapping $\omega \mapsto \tau(\omega)$ is
$\mathbb P^{0}$ almost surely continuous function on 
$C([0, \infty), \mathbb R)$ w.r.t. Skorohod topology. 
(In fact, it's equivalent to
the a.s. continuity w.r.t. max norm on $C([0, T], \mathbb R)$ 
for each $T<\infty$ in this case.) Together with (C2), we have $\tau^{x\_{n}}$ converge to
$\tau^{x}$ in distribution by mapping theorem. 
This shows the continuity of $u$. END.


My question is then, can we have weaker sufficient condition on $\sigma$ to have continuity of $u$?



Sufficient condition of continuity of the expected stopping time
Referring to $X(t)$ and $\tau(x) = \inf \{ t>0 : X(t)<0 \mid X(0) = x \}$ stated above, a Feynman-Kac formula implies that the function $u(x) = \mathbb{E}^x \exp(-\tau)$ satisfies a second-order, linear differential equation: 
$$
\begin{cases}
\frac{1}{2} \sigma(x)^2 u''(x) + u'(x) - u(x) = 0 \\
u(0)=1\;, \quad u(\infty) = 0
\end{cases}
$$ If $\sigma(x)^{-2}$ is integrable, then these equations can be put in the form of a classical Sturm-Liouville problem on a semi-infinite interval. For properties of their solutions see, e.g., Part 4 of Zettl, Anton (2005). *Sturm–Liouville Theory.* Providence: American Mathematical Society. 


