Let $X_t$ be a diffusion process taking values on interval $[l,r]$, and satisfies SDE $dX_t = \mu(X_t)dt + \sigma(X_t)^2/2dB_t$ with $\sigma(\cdot)$ bounded away from 0, and $\mu,\sigma$ bounded. Let $$u(x) = E\left[\left.\int_0^\tau e^{-rt}g(X_t)dt\right|X_0=x\right]$$ be the expected cumulatieve discounted payoff starting from $x$, where $\tau$ is the first exit time from $(a,b)$ for some $l<a<x<b<r$ and $g$ is bounded and measurable.
I want to show that $u\in C^1$.
This is relative easy when $\mu,\sigma$ are Lipschitz continuous. In that case one can write $$u(x) = \int_a^b g(y) G_r(x,y)dy$$ and where $G_r(x,y)$ is the Green function for some boundary value problem of a 2nd-order ODE. The green function can be written as (see Page 19, Handbook of Brownian Motion – Facts and Formulae, 2nd Edition, Borodin and Salminen). $$ G_r(x,y) =\begin{cases} w_r^{-1}\psi_r(x)\phi_r(y), & x\leq y \\ w_r^{-1}\psi_r(y)\phi_r(x), & x\geq y\end{cases} $$ where $\psi_r,\phi_r$ are the fundamental solutions of ODE $$ \sigma(x)^2/2 v''(x) +\mu(x)v'(x)=r v(x). $$ If $\sigma$ and $\mu$ are Lipschitz continuous, then $\psi_r,\phi_r\in C^2$, and the rest of showing $u(x)\in C^1$ is simple (actually in this case one can show $u(x)\in C^2$, see Page 226 of Stochastic calculus: a practical introduction - Durrett).
However in my application $\sigma$ and $\mu$ are piece-wise Lipchitz continuous (let's say has exactly one jump discontinuity point in $(a,b)$).
One way to proceed is to argue that $\psi_r,\phi_r\in C^1$, which is quite obvious in my perspective. Are there any result in the literature stating this fact even if $\sigma$ and $\mu$ is not Lipschitz?
Or does any literature discuss the condition for the smoothness of the cumulated discounted payoff function $u(\cdot)$?