 Self-financing conditions for smooth stock and bond holdings

For $p(x,t)=xg(x,t)+B_{t}h(x,t)$ and deterministic $d{B}_t=\rho B_{t}dt$, Itô formula shows that a self-financing portfolio must have $p_x=g$. This is equivalent to $x{g}_x+B_t{h}_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $x{g}_t+B_t{h}_t+{\sigma }^2{x}^2{g}_x/2=0$. Conversely these two identities give $dp=gdS+hdB$. A value PDE alone does not verify arbitrary prescribed holdings.