Suppose that we have to solve the following optimal control problem
\begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] \end{align}
subject to the constraint
\begin{align} dX^{\alpha}(t) = \mu \alpha dt + \sigma \alpha dW(t) \end{align}
where $t\in[0,T]$, the control $\alpha\in U$ where $U$ is compact, $\mu,\sigma$ are positive constants, $L$, $F$ are real valued functions describing a running cost and a terminal payoff.
My question is how to write the HJB equation for this problem.What i have done till now
\begin{align} V_{t} + \mu \alpha V_{x} + \frac{1}{2}\sigma^{2}\alpha^{2}V_{xx} = 0 \end{align}
i am missing something.