Log-optimal investment with Gaussian drift learning ($w_t^{\ast }=w_0/{\zeta }_t,{\pi }_t^{\ast }=(\sigma m_t-r)/{\sigma }^2$)

For a stock with observed dynamics $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma m_{t}dt$ 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 $1/w_T^{\ast }=\lambda {\zeta }_T$ and the state-price budget constraint give $\lambda =1/w_0$. Replication then gives $w_t^{\ast }=w_0/{\zeta }_t$ 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.