$$\begin{multline}\sup_{(\pi, c, A)}\mathbb E \biggl[\int_0^\infty \!e^{-\beta t}\,u(c_t) \, \mathrm{d}t - \kappa A_0 - \kappa \int_0^\infty \!e^{-\beta t} \, \mathrm{d}A_t \biggr] \\=\sup_{(\pi, c)}\Biggl[\mathbb E \biggl[\int_0^\infty \! e^{-\beta t} \, u(c_t) \, \mathrm{d}t \biggr]-\inf_{A}\mathbb E \biggl[\kappa A_0 + \kappa \int_0^\infty \! e^{-\beta t} \, \mathrm{d}A_t \biggr]\Biggr]\end{multline}$$where $A$ is an increasing continuous process.
I mean, it looks pretty obvious to me, what I wanted to know is if there's some in-between step that I could make explicit. For context, this is an HJB equation coming from an investment-consumption problem with utility $u$ and capital injection process $A$ with cost $\kappa$. Also, $(\pi, c)$ is the amount of money invested-consumption rate that appears in some process that we don't make explicit here because it does not matter.