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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 ${\theta }^{\ast }=-[b(x)J_w+\sigma J_{wx}]/({\sigma }^2{J}_{ww})$. 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.

Use the primitive $U(c)=c^{1-R}/(1-R)$, as in Question 3, and the usual positive spot volatility convention $\sigma >0$. An additive utility function constant shifts both agents' value functions equally and does not affect their policies or the comparison. SetThe Itô formula applied to the stock price and the binary Brownian drift filter givesLet ${\theta }_t$ denote the dollar stock holding. The separated state equations areBoth states use the same observed Brownian motion. In particular, their quadratic covariation is $d[w,X{]}_t=\sigma {\theta }_{t}dt$.

Let $J(w,x)$ be the Bayesian value function. Its Hamilton-Jacobi-Bellman equation isThe cross derivative is necessary: learning and portfolio wealth are correlated through the same observation shock. At a smooth, finite, strictly concave value, the optimal consumption and stock holding areThe number of shares is ${\theta }^{\ast }/S$, and the wealth fraction is ${\theta }^{\ast }/w$.

More explicitly, homogeneity gives $J(w,x)=K(x)w^p/p$, where $p=1-R$. ThenThe additional term is the learning hedge in a binary-drift investment model, a form of intertemporal hedging demand. Merely putting the Bayesian posterior mean drift into a constant-coefficient investment formula would omit it. Eliminating the controls gives the coefficient equationThe economic solution is selected by admissibility and the infinite-horizon value condition, rather than by an arbitrary solution of this ordinary differential equation.

For the comparison, now suppose $0<R<1$ and $\mu >r$. Write $m=\mu -r>0$, $b_{+}=m+\sigma a$, andWhen ${\gamma }_{+}>0$, the informed agent's Merton consumption-investment problem has valueThese are the policies available to an agent who knows the true drift is $+a$.

For any admissible Bayesian policy, not only its optimum, defineThe integration variable is $s$ throughout the accumulated utility function. Applying the Itô formula in the Bayesian observation filtration giveswhereThe informed-agent Hamilton-Jacobi-Bellman equation says ${\sup }_{c,\theta }{H}_{b_{+}}=0$. Maximizing the stock term at any excess drift $b$ givesSince $V_{+}^{\prime \prime }<0$, this maximum increases with $b^2$. The binary Brownian drift filter satisfies $|h(x)|\le a$, and thereforeIt follows that $H_{b(x)}(w,c,\theta )\le {\sup }_{c,\theta }{H}_{b(x)}\le {\sup }_{c,\theta }{H}_{b_{+}}=0$. Thus $Y$ is a local supermartingale. Because $0<R<1$, both accumulated utility function and $V_{+}$ are nonnegative. A nonnegative local supermartingale is a genuine supermartingale, by localization and the conditional Fatou lemma. This proves the requested assertion under the Bayesian probability law, without conditioning that law on the favorable drift.

ConsequentlyThe monotone convergence theorem lets $t$ tend to infinity. Taking the supremum over Bayesian policies proves the maximal squared Sharpe ratio value boundIn particular this holds at the equal-prior state $x=0$. No vanishing-terminal-value assumption is needed for this upper bound, because the terminal term is nonnegative and can simply be dropped.

The finite-value assumption ${\gamma }_{+}>0$ is missing from the printed comparison. Without it, $V_{+}=+\infty$ and the value inequality is trivial, but $Y$ is not a finite supermartingale. For a concrete allowed example, take $r=0$, $\mu =1$, $\sigma =a=1$, $R=1/2$ and $\rho =1$. Then $b_{+}=2$ and ${\gamma }_{+}=-2$. For the informed agent, the constant stock fraction $\pi =4$ and consumption $c=w/2$ giveThe time integral diverges, so his value is infinite. This is why the nontrivial finite-supermartingale proof above needs the stated qualification.

Treat the two-point prior for $\alpha$ as independent of the driving Brownian motion. On a partition $0=t_0<\cdots <t_n=t$, conditionally on $\alpha =\pm a$, the increments of $X$ are independent normal distributions with means $\pm a\Delta t_j$ and variances $\Delta t_j$. Their likelihood ratio isThus every partition containing $t$ gives the same Bayesian posterior, depending only on the endpoint. Refine partitions through a dense set of observation times. Continuity makes their increasing observation fields generate the full path field ${\mathcal{X}}_t$, and convergence of bounded conditional expectations preserves that Bayesian posterior. Equal prior probabilities and the Bayes' theorem consequently giveIts conditional mean drift isThis is the binary Brownian drift filter.

Define the observed innovation processIt is continuous and adapted to ${\mathcal{X}}_t$. For $s<t$, independence of future Brownian motion increments from the past and the latent drift givesFor every $u\ge s$, the tower property of conditional expectation givesSince $|h|\le a$, conditional integration is justified. Subtracting the integrated identities shows $E[{\hat{W}}_t-{\hat{W}}_s\mid {\mathcal{X}}_s]=0$. Thus $\hat{W}$ is a square-integrable martingale. Subtracting a continuous finite variation process does not change quadratic variation, so $[\hat{W}{]}_t=[X{]}_t=[W{]}_t=t$. The Lévy characterization of Brownian motion now proves that $\hat{W}$ is a Brownian motion in the observation filtration. Therefore