= Solution
Put $\kappa=(\mu-r)/\sigma$, assuming $\sigma>0$, and let
$$
\xi_t=\exp\bigl(-rt-\kappa W_t-\tfrac12\kappa^2t\bigr).
$$
This <state-price density> prices the <bank account> and the risky asset. The <Itô formula> gives
$$
d(\xi_tw_t)+\xi_tc_t\,dt=\xi_t(\sigma\theta_t-\kappa w_t)\,dW_t.
$$
Nonnegative wealth and consumption make the deflated cumulative gains a nonnegative <local martingale>, hence a <supermartingale>. Every admissible consumption stream therefore satisfies the <state-price budget constraint> $\mathbb E\int_0^\infty\xi_tc_t\,dt\leq w_0$.
Write $I=(U')^{-1}$ for the <inverse marginal utility>. The <Inada conditions> make $I$ decreasing from infinity to zero. A positive <Lagrange multiplier> $y$ for the budget gives the pointwise condition $e^{-\rho t}U'(c_t)=y\xi_t$. Thus, whenever a finite optimum exists and a multiplier exhausts the budget,
$$
\boxed{c_t^*=I(ye^{\rho t}\xi_t),\qquad\mathbb E\int_0^\infty\xi_tI(ye^{\rho t}\xi_t)\,dt=w_0.}
$$
The <concave supporting-tangent inequality>, integrated against the budget, proves optimality. These curvature and endpoint assumptions alone do not ensure that the infinite-horizon value is finite; part (b) gives the exact additional condition for power utility.
The optimal wealth is the conditional price of remaining consumption:
$$
w_t^*=\xi_t^{-1}\mathbb E\left[\left.\int_t^\infty\xi_sc_s^*\,ds\right|\mathcal F_t\right].
$$
It is nonnegative. Apply the <Martingale representation theorem> to the integrable total discounted consumption claim:
$$
M_t=\mathbb E\left[\left.\int_0^\infty\xi_sc_s^*\,ds\right|\mathcal F_t\right]
=w_0+\int_0^t\phi_s\,dW_s.
$$
Comparing $M_t=\xi_tw_t^*+\int_0^t\xi_sc_s^*ds$ with the deflated wealth equation yields
$$
\boxed{\theta_t^*=\frac1\sigma\left(\frac{\phi_t}{\xi_t}+\kappa w_t^*\right).}
$$
This constructs the portfolio without assuming extra smoothness of the <value function>. If the stationary value $V$ is twice differentiable with $V''<0$, optimization of its <Hamilton-Jacobi-Bellman equation> instead gives $\theta^*(w)=-(\mu-r)V'(w)/(\sigma^2V''(w))$ and $c^*(w)=I(V'(w))$.
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