I'm very new in this field, but I'm giving a try anyways. So please be critical and let me know if something is wrong (We are here to learn so...).
The assertion is not true in general, indeed we have a couple of counterexamples.
A simple counterexample.
Let $\mu\equiv 1$ and $\sigma\equiv 0$. Define
$$A:=(0,1)$$
Then we have for $X_0=1$, the explicit unique solution for $X_t$ given by
$$X_t=1+t$$
In this particular case $$\tau_A=\infty, \ \ \text{ and }\ \ \ \tau_A^-=0.$$
We might add extra conditions to get that $\tau_A=\tau_A^-$ a.s., I guess we can cook up the following simple proposition which is inspired by Saz' comment. I also add some thoughts at the end. But before we do that we need some Lemmas. Let us fix the filtered probability space $(\Omega,\mathcal F,\mathcal F_t,\mathbb P)$.
Lemma 1. For the case $X=B$ we have
$$\tau_A=\tau_A^- \ \ \ \text{ a.s. }$$
Proof. This Lemma is a generalization of what Saz proved here in this answer. The proof should go more or less the same so it will be omitted.
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Now we say the same if we consider time-changed Brownian motions.
Lemma 2. Let $(T(t))_{t\geq 0}$ be a strictly increasing and continuous stochastic process such that $T(0)=0$ and $\lim_{t\to\infty}T(t)=\infty$. Then for $X_t=B_{T(t)}$ we have
$$\tau_A=\tau_A^- \ \ \ \text{ a.s. }$$
Proof. Since $t\mapsto T(t)(\omega)$ for some fixed $\omega\in\Omega$ is increasing and continuous we have the existence of increasing and continuous "inverse random variable" $t\mapsto T^{-1}(t)(\omega)$. The domain of $T$ and $T^{-1}$ (as a function of $t$) is $[0,\infty)$. We have
\begin{align}
\tau_A&=\inf\{t\geq 0 \ :\ B_{T(t)}\in A\}\\
&=\inf\{t\geq 0\ : \ s=T(t), \ B_s\in A\}\\
&=T^{-1}(\inf\{s\geq 0\ : \ B_s\in A\}) \\
&=T^{-1}(\inf\{s\geq 0\ : \ B_s\in \overline{A}\}) \\
&=...\\
&=\tau_A^{-1}
\end{align}
In the third line we have used continuity and monotonicity of $T^{-1}$ and the line before the dots is due to Lemma 1. In the dots we did the steps above with $\overline{A}$ to get the last line.
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Now we can finally state the proposition.
Proposition. Assume that $\mu$ and $\sigma$ are Lipschitz continuous so that a unique solution $(X_t)_{t\geq 0}$ exists for $X_0=x\in\mathbb R$. Moreover assume that $\sigma$ is bounded away from zero, i.e. there is $c>0$ such that $|\sigma|\geq c$. Finally assume that
$$Z_t=\exp\left(-\int^t_0 \frac{\mu(X_s)}{\sigma(X_s)}\,dB_s+\frac 1 2 \int^t_0\frac{\mu(X_s)^2}{\sigma(X_s)^2}\,ds \right)$$
is a $\mathcal F_t$-martingale. Then we have that $\tau_A=\tau_A^-$ a.s..
Proof. Sinze $Z_t$ is a martingale, we can define a probability measure $\mathbb Q$, that is equivalent to $\mathbb P$, by
$$\frac{d\mathbb Q}{d\mathbb P}|_{\mathcal F_t}=Z_t$$
Applying Girsanov gives us that
$$W_t=B_t+\int^t_0\frac{\mu(X_s)}{\sigma(X_s)}\,ds$$
is a BM w.r.t. $\mathbb Q$. We also verify w.r.t. $\mathbb Q$ we can write $X$ as
$$X_t=x+\int^t_0\sigma(X_s) \,dW_s$$
Now we notice that
$$\langle X\rangle_t=\int^t_0 \sigma(X_s)^2\,ds\geq \int^t_0 c^2\,ds=tc^2$$
Therefore $\langle X\rangle_\infty=\infty$. By the Theorem of Dubins-Schwarz, there exists a BM w.r.t. $\mathbb Q$, say $\tilde W$, such that
$$X_t=x+\tilde W_{\langle X\rangle_t}$$
Now take $T(t)=\langle X\rangle_t$. Apply Lemma 2 with this $T(t)$ and $A'=A-\{x\}$ to get the proof done.
Some remarks/thoughts.
- The proposition shows that, in particular, a BM with drift satisfies the property.
- There are lot of ways to show that $Z_t$ is a martingale, maybe Novikov's or Kazamaki are applicable.
- Thanks to Saz who told me to use Dubins-Schwarz Theorem to mke the proposition applicable to a larger class of Ito-diffusions.
