= Log-optimal investment with Gaussian drift learning
{title2=$w_t^*=w_0/\zeta_t,\quad\pi_t^*=(\sigma m_t-r)/\sigma^2$}
For a <stock> with observed dynamics $dS_t/S_t=\sigma d\widehat W_t+\sigma m_tdt$ and constant riskless <interest rate> $r$, the <finite-horizon pricing density for Gaussian drift learning> with $k=r/\sigma$ gives $\zeta_t=e^{-rt}D_t$. The terminal <logarithmic utility> condition $1/w_T^*=\lambda\zeta_T$ and the <state-price budget constraint> give $\lambda=1/w_0$. Replication then gives $w_t^*=w_0/\zeta_t$ at every time. The <Itô formula> for $1/\zeta$ makes the optimal dollar stock fraction $(m_t-r/\sigma)/\sigma$. Its finite expected log return follows from finite posterior second moments on finite horizons.
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