In reading [this post][1] I couldn't help but wonder the following question:

Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem:
$$
\begin{aligned}
v(t,x)= & sup_{a \in \mathcal{A}} \, \mathbb{E}^{t,x}\left[
\int_t^T f(X_s^a,a_s)ds + g(X_T^a)
\right]\qquad 0\leq t< T
\\
dX_t^a =&  x_0 + \int_0^t b(X_s^a,a_s)ds + \int_0^t \sigma dW_s,
\end{aligned}
$$
optimized over the set of all predictably measurable controls.  In this case, we know that (under some conditions so that BSDE methods apply) there exist exists an optimal control $\hat{a}_t$ and it is of the form:
$$
a_t = \alpha(X_t,V_t,\nabla V_t,\Delta V_t),
$$
where $V_t$ is the problem's value function (which is also a local-martingale).  

My question is, under what (reasonable) conditions is:

 - $\alpha$ continuous,
 - $V_t,\nabla V_t$, and $\Delta V_t$ can all be expresses as strong solutions to *some* (finite-dimensional) time-homogeneous SDEs?

  [1]: https://mathoverflow.net/questions/348292/confusion-optimal-control-abuse-notation?noredirect=1#comment872348_348292