= Square-root investment-consumption value
{title2=$V(x,t)=2\sqrt{x}\sqrt{h(t)}$}
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 $h'+qh+1=0$, $h(T)=1$, 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.
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