In reading this post 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?