-\frac{\partial V}{\partial t}(x, t) = \min_{u} \left\{ L(x, u, t) + \nabla_x V(x, t)^T f(x, u, t) + \frac{1}{2} \text{Tr}\left( \sigma(x, t) \sigma(x, t)^T \nabla_{xx}^2 V(x, t) \right) \right\}