Logarithmic spending with a terminal multiplier ($u^{\ast }(t;\mu )={\mu }^{-1}{e}^{(\beta -r)(T-t)}$)

For bank dynamics $\dot{x}=A+rx-u$ and utility ${\int }_0^T{e}^{-\beta t}\log udt+\mu e^{-\beta T}x(T)$, substituting the terminal balance separates the concave optimization pointwise and gives the displayed control for $\mu >0$. Put $W=e^{rT}x(0)+A(e^{rT}-1)/r$ and $B={\int }_0^T{e}^{\beta s}ds$. Its terminal balance is $W-B/\mu$, so the unique positive multiplier achieving zero terminal wealth is $B/W$. The same control solves the problem with no terminal reward and constraint $x(T)\ge 0$, by the supporting-line inequality for logarithms.