= Solution
Use $\theta_t$ for the dollar amount invested in the <stock>; the number of shares is $\theta_t/S_t$. The <bank account> contains $w_t-\theta_t$ dollars. With no intermediate <consumption>, the <self-financing portfolio> equation is
$$
\boxed{dw_t=r_t(w_t-\theta_t)dt+\theta_t\frac{dS_t}{S_t}
=[r_tw_t+\theta_t(\mu_t-r_t)]dt+\theta_t\sigma_tdW_t.}
$$
If instead $\theta$ denotes share holdings, replace each dollar $\theta_t$ here by $S_t\theta_t$. The equation says that trading only reallocates existing <portfolio wealth>; gains come from cash interest and stock returns.
For the standard unconstrained complete-diffusion formulation, assume the completed <natural Brownian filtration>, initial wealth $w_0>0$, and nonnegative admissible wealth. Define the <market price of risk> $\kappa_t=(\mu_t-r_t)/\sigma_t$. The boundedness assumptions make $\kappa$ bounded. The <state-price density> normalized at zero is
$$
\boxed{\zeta_t=\exp\left(-\int_0^t r_sds-\int_0^t\kappa_sdW_s-\frac12\int_0^t\kappa_s^2ds\right),\qquad d\zeta_t=-\zeta_t(r_tdt+\kappa_tdW_t).}
$$
Its money-market-adjusted process $Z_t=e^{\int_0^t r_sds}\zeta_t$ is a true <martingale> by the <Novikov condition>. Under $d\mathbb Q=Z_Td\mathbb P$, the <Girsanov theorem> gives $W_t^{\mathbb Q}=W_t+\int_0^t\kappa_sds$ as a <Brownian motion>, and the discounted stock has zero drift.
The <Itô product rule> applied to the wealth equation gives
$$
d(\zeta_tw_t)=\zeta_t(\sigma_t\theta_t-\kappa_tw_t)dW_t.
$$
For nonnegative admissible wealth this is a <nonnegative local martingale>, hence a <supermartingale>. Therefore every affordable terminal payoff $X=w_T$ satisfies the <state-price budget constraint> $\mathbb E[\zeta_TX]\leq w_0$. Nonzero volatility and the <Brownian martingale representation theorem> allow replication of nonnegative claims with finite state-price cost.
For a differentiable strictly <concave> <utility function> with the usual conditions ensuring an interior integrable optimizer, put $I=(U')^{-1}$, the <inverse marginal utility>. Pointwise maximization of $U(x)-y\zeta_Tx$ gives
$$
\boxed{w_T^*=I(y\zeta_T),\qquad \mathbb E[\zeta_TI(y\zeta_T)]=w_0.}
$$
The second equation determines the positive constant $y$. To prove optimality, use the <concave supporting-tangent inequality>:
$$
U(X)-U(w_T^*)\leq U'(w_T^*)(X-w_T^*)=y\zeta_T(X-w_T^*).
$$
Taking expectations and using the budget bound proves that the candidate dominates every admissible affordable payoff, whenever the utility expectations are well defined. This is the <complete-market terminal utility optimizer>.
The actual wealth and dollar <portfolio> are as explicit as general adapted coefficients permit. Let
$$
N_t=\mathbb E[\zeta_TI(y\zeta_T)\mid\mathcal F_t],\qquad dN_t=h_tdW_t.
$$
The <Brownian martingale representation theorem>, with localization if only $L^1$ integrability is initially known, gives $h$. The <dollar portfolio from a deflated wealth martingale> is
$$
\boxed{w_t^*=\frac{N_t}{\zeta_t},\qquad
\theta_t^*=\frac{h_t/\zeta_t+\kappa_tw_t^*}{\sigma_t}.}
$$
The bank holds $w_t^*-\theta_t^*$ dollars. Equivalently, with $B_t=e^{\int_0^t r_sds}$, $w_t^*=B_t\mathbb E^{\mathbb Q}[w_T^*/B_T\mid\mathcal F_t]$. These formulas need no unjustified differentiability of a value function with random coefficients.
The printed assumption that $U$ is only increasing and concave does not guarantee that the interior formula exists, or even that the optimal value is finite. In general replace the inverse formula by an affordable selection
$$
X^*(\omega)\in\operatorname*{arg\,max}_{x\in\mathcal D}\{U(x)-y\zeta_T(\omega)x\},
$$
where $\mathcal D$ is the wealth domain; boundary choices or supporting slopes handle nondifferentiability. Existence, finite cost and replication still have to hold. A concrete obstruction is the <unbounded linear terminal-wealth utility>: take $U(x)=x$, $r=0$, $\mu=\sigma=1$, and the Brownian filtration. Then $\zeta_T=e^{-W_T-T/2}$. For $A_n=\{\zeta_T<1/n\}$, the bounded nonnegative claim
$$
X_n=\frac{w_0\mathbf1_{A_n}}{\mathbb E[\zeta_T\mathbf1_{A_n}]}
$$
is replicable, costs $w_0$, and has $\mathbb EX_n>nw_0$. Thus there is no finite optimal expected utility under the literal utility hypotheses alone. Also, bounded coefficients do not ensure <market completeness> in an arbitrarily enlarged filtration containing untraded randomness; the full replication formula uses the Brownian market information assumption stated above.
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