= Learning hedge in a binary-drift investment model
{title2=$\pi^*(x)=[b(x)+\sigma K'(x)/K(x)]/(R\sigma^2)$}
With the <binary Brownian drift filter>, a stock of observed excess drift $b(x)=\mu-r+\sigma a\tanh(ax)$ is driven by the same <Brownian motion> as its information state $X$. The <Hamilton-Jacobi-Bellman equation> for <portfolio wealth> $w$, dollar stock holding $\theta$ and value $J(w,x)$ includes $\sigma\theta J_{wx}$. Consequently $\theta^*=-[b(x)J_w+\sigma J_{wx}]/(\sigma^2J_{ww})$. For <CRRA utility>, <homogeneity> gives $J=K(x)w^{1-R}/(1-R)$ and hence the displayed stock fraction. The second term is <intertemporal hedging demand>; replacing the unknown drift by its posterior mean while omitting this term does not generally give the optimal policy.
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