= Maximal squared Sharpe ratio value bound
{title2=$|b_t|\leq b_+\ \Longrightarrow\ J(w)\leq V_+(w)$}
For $0<R<1$, common constant interest and volatility, and observed excess drift bounded in absolute value by $b_+>0$, let $V_+$ be the finite <Merton consumption-investment problem> value for excess drift $b_+$. In its <Hamilton-Jacobi-Bellman equation>, optimizing the stock holding contributes $-b^2(V_+')^2/(2\sigma^2V_+'')$. Since $V_+''<0$, this contribution increases with $b^2$. Thus every admissible policy makes accumulated discounted <CRRA utility> plus discounted $V_+$ a nonnegative <local supermartingale>, hence a <supermartingale>. Dropping its nonnegative terminal term and using the <monotone convergence theorem> proves the value bound. The condition $[\rho-(1-R)(r+b_+^2/(2R\sigma^2))]/R>0$ is essential for the finite supermartingale construction.
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