= Solution
\b[The joint generator.] With $\theta$ again measured in dollars, <portfolio wealth> obeys
$$
dw=\sigma(x)\theta\,dW+[rw+(\mu(x)-r)\theta-c]\,dt.
$$
Its noise and the factor noise are driven by the same <Brownian motion>. Their <quadratic covariation> is $\sigma(x)\theta\alpha(x)\,dt$, so the <diffusion generator> has a cross derivative. <Dynamic programming> and the <Itô formula> give
$$
\boxed{\begin{aligned}
0={}&\tfrac12\alpha^2V_{xx}+\beta V_x-\rho V+rwV_w\\
&+\sup_{\theta\in\mathbb R}
\left\{(\mu-r)\theta V_w+\sigma\alpha\theta V_{wx}
+\tfrac12\sigma^2\theta^2V_{ww}\right\}
+\sup_{c\geq0}\{U(c)-cV_w\}.
\end{aligned}}
$$
All coefficient functions in this formula are evaluated at $x$. The cross derivative is essential: it gives <intertemporal hedging demand>. Normalizing <CRRA utility> as $U(c)=c^p/p$, where $p=1-R$, the two optimizations give, at nonzero volatility,
$$
c^*=(V_w)^{-1/R},\qquad
\theta^*=-\frac{(\mu-r)V_w+\sigma\alpha V_{wx}}{\sigma^2V_{ww}},
$$
and hence
$$
0=\tfrac12\alpha^2V_{xx}+\beta V_x-\rho V+rwV_w
-\frac{[(\mu-r)V_w+\sigma\alpha V_{wx}]^2}{2\sigma^2V_{ww}}
+\frac R p(V_w)^{p(-1/R)}.
$$
Here $p(-1/R)=1-1/R$.
\b[Homogeneity and the reduced equation.] Scaling initial <portfolio wealth> and both controls preserves the <portfolio wealth> constraint and multiplies the objective by the positive number $b^p$. Thus
$$
V(w,x)=\frac{w^p}{p}f(x),\qquad f(x)>0.
$$
This expression is valid for both signs of $p$: $V$ is negative when $R>1$, but its <portfolio wealth> derivative is positive. Its derivatives are
$$
V_w=w^{-R}f,\quad V_{ww}=-Rw^{-R-1}f,\quad
V_{wx}=w^{-R}f',\quad
V_x=\frac{w^p}{p}f',\quad V_{xx}=\frac{w^p}{p}f''.
$$
Substitution yields
$$
\boxed{
\tfrac12\alpha^2f''+\beta f'+(pr-\rho)f
+\frac{p[(\mu-r)f+\sigma\alpha f']^2}{2R\sigma^2f}
+R f^{(R-1)/R}=0.}
$$
The resulting feedback is
$$
\boxed{\frac{c^*}{w}=f^{-1/R},\qquad
\frac{\theta^*}{w}=\frac{\mu-r}{R\sigma^2}
+\frac{\alpha}{R\sigma}\frac{f'}f.}
$$
The first portfolio term is myopic, and the second is <intertemporal hedging demand>.
\b[Constant market price of risk.] If $\mu-r=\sigma\kappa$ and volatility is nonzero, the portfolio term becomes $p(\kappa f+\alpha f')^2/(2Rf)$, so the magnitude of stock volatility disappears. Applying the <power transformation of a complete-market investment equation> $f=g^R$ cancels the squared-gradient terms and gives the further reduction
$$
\boxed{\tfrac12\alpha^2g''+
\left(\beta+\frac{p\kappa\alpha}{R}\right)g'
-\gamma_M g+1=0,\qquad
\gamma_M=\frac{\rho-p(r+\kappa^2/(2R))}{R}.}
$$
For $\gamma_M>0$ its economic solution is $g=1/\gamma_M$. Thus
$$
\boxed{V(w,x)=\frac{\gamma_M^{-R}w^p}{p},\qquad
c^*=\gamma_M w,\qquad
\theta^*=\frac{\kappa w}{R\sigma(x)}.}
$$
To see why this solves the <investment-consumption problem>, optimize directly over <Brownian portfolio exposures>, writing $y=\sigma(x)\theta$. The <portfolio wealth> equation becomes $dw=y\,dW+(rw+\kappa y-c)dt$, which no longer contains $X$. With nonzero volatility the same admissible exposure processes are available for every factor state, so the attainable wealth-consumption pairs, and therefore the value, are exactly those of the <Merton consumption-investment problem>. This also excludes extraneous solutions of the linear equation without imposing artificial factor boundary data.
The printed boundedness assumptions do not ensure nonzero volatility or a finite value. The unsimplified <HJB equation> remains the correct control equation at a zero of $\sigma$. There the hedge term vanishes; if $\mu\ne r$, the riskless excess return gives <arbitrage> with unrestricted holdings. Under $\mu-r=\sigma\kappa$, a zero-volatility state offers only the bank exposure at that instant. For example $\sigma\equiv0$, $\mu\equiv r$ satisfies this relation for any chosen $\kappa$, but its value uses $\gamma_0=[\rho-pr]/R$, not a fictitious nonzero risk premium. \b[The constant-value formula using $\kappa$ presupposes access to the Brownian exposure], with sufficient integrability for the corresponding holdings. Additive utility constants again only shift $V$ by a constant divided by $\rho$.
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