= Solution
Assume, as in the Bayesian model, that the unknown drift $\alpha$ is independent of the driving <Brownian motion>, with known positive prior precision $\tau_0$. Recover the observed signal from the positive <stock> price:
$$
Y_t=\frac1\sigma\log(S_t/S_0)+\frac\sigma2t=\alpha t+W_t.
$$
For a fixed time $t$, the centered observed path $Y_s-(s/t)Y_t$ is a <Brownian bridge> independent of $(\alpha,Y_t)$: it is independent of the prior and has zero covariance with $W_t$ in their joint <Gaussian> law. Thus the whole observed history has the same posterior information about $\alpha$ as its endpoint. This proves <Brownian endpoint sufficiency for a constant drift> rather than assuming it.
The likelihood of that endpoint, as a function of $\alpha$, is proportional to $\exp(\alpha Y_t-\alpha^2t/2)$. Multiplying by the <normal distribution> prior and completing the square gives the <Gaussian Brownian drift filter>
$$
\boxed{\alpha\mid\mathcal F_t^S\sim N(\widehat\alpha_t,\tau_t^{-1}),\qquad
\tau_t=\tau_0+t,\qquad \widehat\alpha_t=\frac{\tau_0\widehat\alpha_0+Y_t}{\tau_0+t}.}
$$
Define the observed <innovation process> $\widehat W_t=Y_t-\int_0^t\widehat\alpha_sds$. It is a continuous observed-filtration <martingale>: for $s<u$, independence of future noise and the <tower property of conditional expectation> give $\mathbb E[\alpha-\widehat\alpha_u\mid\mathcal F_s^S]=0$, so the conditional mean of its increment vanishes. Its <quadratic variation> is $t$, since subtracting the finite-variation drift from $Y$ does not change quadratic variation. The <Lévy characterization of Brownian motion> makes $\widehat W$ an observed-filtration <Brownian motion>. Differentiating the posterior mean gives
$$
\boxed{d\widehat\alpha_t=\tau_t^{-1}(dY_t-\widehat\alpha_tdt)=\tau_t^{-1}d\widehat W_t.}
$$
Conversely, the last equation determines $\widehat\alpha$ from $\widehat W$ with deterministic coefficients and known initial value, and $Y=\widehat W+\int\widehat\alpha\,dt$. Thus the observed and innovation <filtrations> coincide, supplying the natural Brownian information needed for replication. Consequently the observed stock dynamics are $dS_t/S_t=\sigma d\widehat W_t+\sigma\widehat\alpha_tdt$. Its observed <market price of risk> is $\kappa_t=\widehat\alpha_t-r/\sigma$, and its normalized <state-price density> solves
$$
d\zeta_t=-\zeta_t(rdt+\kappa_td\widehat W_t),\qquad
\zeta_t=\exp\left(-rt-\int_0^t\kappa_sd\widehat W_s-\frac12\int_0^t\kappa_s^2ds\right).
$$
The posterior mean is unbounded, so bounded-coefficient <Novikov condition> reasoning from Question 2 is not available automatically. A direct finite-horizon argument supplies true pricing. Relative to driftless <Wiener measure> for $Y$, the Bayesian mixture has observation density
$$
L_t(Y_t)=\sqrt{\frac{\tau_0}{\tau_0+t}}\exp\left(\frac{(\tau_0\widehat\alpha_0+Y_t)^2}{2(\tau_0+t)}-\frac{\tau_0\widehat\alpha_0^2}{2}\right).
$$
A law with observation drift $k=r/\sigma$ instead has density $e^{kY_t-k^2t/2}$. Therefore
$$
\boxed{D_t=\frac{e^{kY_t-k^2t/2}}{L_t(Y_t)},\qquad \zeta_t=e^{-rt}D_t.}
$$
This is the <finite-horizon pricing density for Gaussian drift learning>. The ratio changes the physical observed law into a genuine probability law, hence has mean one. Indeed, differentiation of the explicit likelihood, using $dY_t=d\widehat W_t+\widehat\alpha_tdt$, gives
$$
d\log L_t=\widehat\alpha_t\,dY_t-\frac12\widehat\alpha_t^2dt,\qquad
d\log D_t=(k-\widehat\alpha_t)d\widehat W_t-\frac12(\widehat\alpha_t-k)^2dt.
$$
The <Itô formula> therefore gives $dD_t=-D_t\kappa_td\widehat W_t$, agreeing with the stochastic exponential above. Under this pricing law the stock drift is $\sigma k=r$.
For <logarithmic utility>, terminal marginal utility is $1/w_T$. The <state-price budget constraint> and the marginal condition imply $w_T^*=1/(y\zeta_T)$ and $1/y=w_0$. Since $\zeta_Tw_T^*=w_0$ is constant, its conditional-expectation pricing process is constant as well. Thus the entire optimal wealth process is
$$
\boxed{w_t^*=\frac{w_0}{\zeta_t}.}
$$
Indeed, for any other nonnegative admissible terminal wealth $X$, the elementary inequality $\log(X/w_T^*)\leq X/w_T^*-1$ gives
$$
\mathbb E\log(X/w_T^*)\leq\mathbb E[\zeta_TX/w_0]-1\leq0.
$$
This directly verifies optimality. Expected candidate log wealth is finite because $\mathbb E\widehat\alpha_t^2=\widehat\alpha_0^2+\tau_0^{-1}-(\tau_0+t)^{-1}$ and its time integral is finite on every fixed horizon. Applying the <Itô formula> to $1/\zeta$ gives
$$
\frac{dw_t^*}{w_t^*}=(r+\kappa_t^2)dt+\kappa_td\widehat W_t.
$$
The observed <self-financing portfolio> with risky dollar fraction $\pi_t$ has diffusion coefficient $\pi_t\sigma$. Matching the coefficients proves the <log-optimal investment with Gaussian drift learning> rule
$$
\boxed{\pi_t^*=\frac{\kappa_t}{\sigma}=\frac{\sigma\widehat\alpha_t-r}{\sigma^2},\qquad\theta_t^*=\pi_t^*w_t^*.}
$$
The remaining dollars are held in the <bank account>; the fraction may be negative or exceed one because the question imposes neither short-sale nor borrowing restrictions.
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