= Solution
\b[Wealth and admissibility.] The dollar holding $\theta$ earns the risky return, while $w-\theta$ earns the <continuous-time bank account> return. Removing <consumption> therefore gives
$$
\boxed{dw_t=\sigma\theta_t\,dW_t+\bigl[rw_t+(\mu-r)\theta_t-c_t\bigr]dt.}
$$
Here $\theta$ is a dollar amount, not a number of shares; the number of shares is $\theta_t/S_t$. Both controls must use available information: take $\theta$ <predictable> and $c\geq0$ <progressively measurable>, with
$$
\int_0^t(\sigma\theta_s)^2ds<\infty,\qquad
\int_0^t\bigl(|(\mu-r)\theta_s|+c_s\bigr)ds<\infty
$$
almost surely on every finite interval. Require a well-defined objective and an <admissible trading strategy> satisfying $w_t\geq0$. At zero <portfolio wealth> this excludes continued risky gambling or positive <consumption>. The <state-price budget constraint> rules out doubling strategies. Throughout the diffusion calculations take $\sigma\ne0$, positive discount $\rho$, positive decay $\lambda$, and finite value; degeneracies are discussed where they affect the conclusions.
\b[Satisfaction and dynamic programming.] Write the <consumption satisfaction stock> as
$$
\xi_t=e^{-\lambda t}\left(\xi_0+\int_0^t e^{\lambda s}c_sds\right).
$$
The <product rule> gives, almost everywhere in time,
$$
\boxed{d\xi_t=(c_t-\lambda\xi_t)\,dt.}
$$
This state is a <finite-variation process>, so it has no <quadratic covariation> with <portfolio wealth>. Applying the <Itô formula> to the discounted <value function> over a short interval, then using <dynamic programming>, gives the interior <HJB equation>
$$
0=U(\xi)-\rho V+rwV_w-\lambda\xi V_\xi
+\sup_{\theta\in\mathbb R}\left\{(\mu-r)\theta V_w+\frac{\sigma^2\theta^2}{2}V_{ww}\right\}
+\sup_{c\geq0}c(V_\xi-V_w).
$$
Assuming $V_{ww}<0$, completing the square gives
$$
\theta^*=-\frac{\mu-r}{\sigma^2}\frac{V_w}{V_{ww}},
\qquad
\sup_\theta\{\cdots\}=-\frac{\kappa^2}{2}\frac{V_w^2}{V_{ww}},
\qquad
\kappa=\frac{\mu-r}{\sigma}.
$$
The last supremum is zero if $V_\xi-V_w\leq0$ and infinite otherwise. Thus \b[the <gradient constraint for unbounded consumption> is $V_\xi\leq V_w$, and $c=0$ wherever the inequality is strict.] Since there is no direct penalty for a very large <consumption> rate, an active boundary can involve <singular consumption control>. In that relaxed interpretation the <HJB equation> is
$$
\max\left\{U(\xi)-\rho V+rwV_w-\lambda\xi V_\xi
-\frac{\kappa^2}{2}\frac{V_w^2}{V_{ww}},\ V_\xi-V_w\right\}=0.
$$
With ordinary rate controls, this describes the supremum and its limiting transfer policy; it does not promise that an instantaneous transfer is attained by a finite rate.
\b[Power reduction.] An additive constant in the <utility function> only adds a control-independent constant divided by $\rho$ to the value, so normalize $U(\xi)=\xi^{1-R}/(1-R)$. Put $p=1-R$ and $x=w/\xi$ for $\xi>0$. Scaling <portfolio wealth>, <consumption satisfaction> and the controls by the same positive number gives the <wealth-to-satisfaction reduction>
$$
V(\xi,w)=\xi^p v(x),\qquad
V_w=\xi^{-R}v',\quad
V_{ww}=\xi^{-R-1}v'',\quad
V_\xi=\xi^{-R}(pv-xv').
$$
Consequently the reduced <HJB equation> is
$$
\boxed{\max\left\{
\frac1p-[\rho+\lambda p]v+(r+\lambda)xv'
-\frac{\kappa^2}{2}\frac{(v')^2}{v''},
\ pv-(x+1)v'
\right\}=0.}
$$
In the strict waiting region the first expression vanishes and <consumption> is zero. On a transfer region the second vanishes; integrating it gives $v(x)=K(1+x)^p$. This reflects preservation of $w+\xi$ during an instantaneous wealth-to-satisfaction transfer.
\b[Why a waiting threshold is expected, and its qualification.] The <gradient constraint for unbounded consumption> compares the benefit of increasing <consumption satisfaction> with the opportunity cost of spending financial <portfolio wealth>. When <consumption satisfaction> is already large relative to cash, waiting lets <consumption satisfaction> decay while financial <portfolio wealth> earns returns; consuming immediately can be wasteful. <Homogeneity> makes the comparison depend only on $x$. For fixed $s=w+\xi$, joint <concavity> of the <value function> makes $q_s(a)=V(s-a,a)$ <concave> in financial <portfolio wealth> $a$. It is nondecreasing because an immediate transfer can reproduce any smaller financial allocation. Consequently its derivative $V_w-V_\xi$ is nonnegative and nonincreasing in $a$: a strict waiting region, if present, starts at zero financial <portfolio wealth> and ends at a single transfer boundary. <Homogeneity> makes the corresponding boundary a ratio $x_*$. In the usual finite-boundary regime this gives $c=0$ for $x<x_*$, and transfers push a larger ratio down towards $x_*$. The condition is a comparison of marginal values, not the ordinary formula $c=(V_w)^{-1/R}$: current <utility function> here depends on <consumption satisfaction>, not on current <consumption>.
A positive threshold is not guaranteed by the printed hypotheses alone. A useful sufficient local test illustrates the intended argument. Let $a=\rho+\lambda p>0$. At zero <portfolio wealth>,
$$
V(\xi,0)=\frac{\xi^p}{pa},\qquad V_\xi(\xi,0)=\frac{\xi^{-R}}a.
$$
Starting with a small extra <portfolio wealth> $\varepsilon$, holding it in the <continuous-time bank account> until a fixed time $t$, then transferring it into <consumption satisfaction>, has right derivative in $\varepsilon$ at zero equal to
$$
\frac{\xi^{-R}}a\,e^{(r-\rho+\lambda R)t}.
$$
Therefore, if $\rho<r+\lambda R$, the <portfolio wealth> marginal value is strictly larger than the <consumption satisfaction> marginal value at zero. With the usual continuity of marginal values, there is a positive interval on which the <gradient constraint for unbounded consumption> is strict. The fixed-total-resource <concavity> argument then gives the threshold structure.
For a concrete counterexample to an unconditional positive threshold, take $R=1/2$, $\lambda=1$, $r=\mu=1/10$, $\sigma=1$ and $\rho=2$. There is zero <market price of risk>. The relaxed value is
$$
F(\xi,w)=\frac{(\xi+w)^p}{pa},\qquad p=\frac12,\quad a=\frac52.
$$
Indeed $F_\xi=F_w$, $F_{ww}<0$, and the optimized waiting residual, after division by $\xi^p$, is
$$
-\int_0^x(1+s)^{-R}ds+\frac{r+\lambda}{a}x(1+x)^{-R}\leq0,
$$
because $(r+\lambda)/a<1$ and the integrand is decreasing. The <Itô formula> gives an upper bound by $F$, while transferring all <portfolio wealth> into <consumption satisfaction> over intervals tending to zero attains that bound in the limit. Hence the transfer boundary is $x_*=0$ in this example. \b[The positive-threshold explanation needs a parameter regime supporting a genuine waiting region.] For $R>1$, even the zero-wealth value is finite only if $\rho>\lambda(R-1)$; otherwise decaying <consumption satisfaction> gives value $-\infty$.
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