Graduate Smoothness of a value function with discontinuous parameters

[economicsnerd]

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).

I'm interested in the function $v:\mathbb{R}\to\mathbb{R}$ given by $v(x) = \mathbb E \left[ \int_0^\infty e^{-\int_0^t r(X_\tau) d\tau} f(X_t) \bigg| X_0 = x, \ dX_t = \mu(X_t) dt + dB_t \right]$, where $\{B_t\}_t$ is a standard Brownian motion.

I know that $v$ is well-defined, and I'm confident that it's continuous. I'm wondering if I can get any stronger smoothness results than this. For instance, is $v$ continuously differentiable?

 

[economicsnerd]

Related question:

If everything above were $C^2$, then $v$ would satisfy the DE, $r(x)v(x) = f(x) + \mu(x)v'(x) + \tfrac12 v''(x)$. I'd love to know if, when $\mu, f, r$ are discontinuous, there's still any meaningful sense in which the DE is satisfied.