consider the Bellman Equation
\begin{equation*} V(\alpha)=\max_{\beta} f(\beta,\alpha)+A(\beta,\alpha) V(\alpha)+B(\beta,\alpha) V'(\alpha) \end{equation*}
How can I show the existence of the solution? I tried to translate it into an Optimal Control one and use Pontryagin, but I don't think I can do so...