Let $\bar{x}, \underline{x}: [0,\infty)\rightarrow \mathbb{R}$ be continuous functions, $\bar{x}(t) > \underline{x}(t)$ for all $t \geq 0$ and define $U := \{(t,x) | t \in (0,\infty), x\in [\underline{x}(t), \bar{x}(t)]\}$. Let $V : [0,T] \rightarrow \mathbb{R}$ be some differentiable function.
Consider a standard Brownian motion $(W_t)_{t \geq 0}$ and for any $x \in [\underline{x}(0), \bar{x}(0)]$ let $\tau := \inf[t > 0| (t,W_t) \notin U]$ be the first hitting time of when the process $W_t$ first hit one of the boundary $x = \bar{x}(t)$ or $x = \underline{x}(t)$.
My question: is the function $f(t,x) := \mathbb{E}[V(\tau)|W_t=x] \in C^{1,2}$ on $U$ (i.e. at least once and twice continuously differentiable in $t$ and $x$, respectively)?
I'd like to be able to properly use Ito's lemma to write down a PDE $\partial_t f + \frac{1}{2}\partial_x^2 f = 0$ later. I believe this is probably fine for constant $\underline{x}$ and $\bar{x}$, or even smooth $\underline{x}$ and $\bar{x}$, but I'd like to also know when they are just continuous. Afterall, we are considering a diffusion of something, so I hope the expectation above would be smooth, but I'm not sure how to proceed mathematically.