Suppose $b : \mathbb{R} \to \mathbb{R}$ and $\sigma: \mathbb{R}\to \mathbb{R}$ are Lipschitz and that $(X_t)_{t\ge0}$ is a diffusion with $X_0 = x_0$ and $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ .
Consider the PDE $$b(x)v'(x) + \frac{\sigma(x)^2}{2}v''(x) = 0.$$
If $v$ is a classical supersolution to the problem, i.e. $v$ is twice differentiable and $$b(x)v'(x) + \frac{\sigma(x)^2}{2}v''(x) \le 0,$$ then $(v(X_t))_{t\ge 0}$ is a local supermartingale, by Ito's lemma.
In the case that $v$ is a lower semicontinuous viscosity supersolution, does this still hold?
