# Smoothness of a value function with discontinuous parameters

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?

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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.

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