We interpret the following fully nonlinear second order partial differential equation $$\left\{ \begin{gathered}\partial _t u + \mathop {\inf }\limits_\alpha \left\{ {\mathcal{L}\left( {x, \alpha } \right)u + f\left( {x, u, \partial _x u\sigma \left( {x, \alpha } \right),\alpha } \right)} \right\} = 0,\left( {x, t} \right) \in D \times \left( {0, T} \right), \hfill \\for \left( {x, t} \right) \in D \times \left[ {0, T} \right];u\left( {x, T} \right) = g\left( x \right). \hfill \\\end{gathered} \right.$$ as the value function of certain optimal controlled diffusion problem. Where A ∈ ℝk is control domain. xxxL(x, α) is a second order elliptic partial differential operator parametrized by the control variable α ∈ A ⊂ ℝk. A particular case of this equation is when f=f(x, α). In this case, the equation is the well known Hamilton Jacobi Bellman equation.