Square-root investment-consumption value ($V(x,t)=2\sqrt{x}\sqrt{h(t)}$)

For zero cash interest, constant excess drift $\mu$, volatility $\sigma$, and both terminal and consumption utility $2\sqrt{x}$, set $A=\sigma {\sigma }^T$ and assume $\mu$ belongs to the image of a linear map $A$. Let $A^{+}$ be the Moore-Penrose inverse and $q={\mu }^T{A}^{+}\mu$. Then $h^{\prime }+qh+1=0$, $h(T)=1$, so $h(t)=e^{q(T-t)}+(e^{q(T-t)}-1)/q$ for $q>0$, and $h(t)=1+T-t$ for $q=0$. The Hamilton-Jacobi-Bellman equation has the displayed solution. Maximizing dollar investment and consumption are ${\theta }_t=2X_t{A}^{+}\mu$ and $C_t=X_t/h(t)$. The resulting positive portfolio wealth is a geometric Brownian motion with a deterministic time-dependent drift, and finite-horizon moment bounds justify equality in verification.