= Absorption-time bound from conditional variance and overshoot
{title2=$\mathbb E(T\wedge T_R)\leq(\tau R-1)/\sigma^2$}
Let a nonnegative <martingale> start at one, let $T$ be its first zero, and let $T_R$ be its first hit of $[R,\infty)$. Suppose $\mathbb E[(M_{n+1}-M_n)^2\mid\mathcal F_n]\geq\sigma^2\mathbf1_{\{T>n\}}$ and $M_{T_R}\leq\tau R$ on finite hits, for thresholds $R\geq1$. Then $M_{T\wedge T_R\wedge n}^2-\sigma^2(T\wedge T_R\wedge n)$ is a <submartingale>. The bounded stopped values have mean one, so their second moment is at most $\tau R$. Consequently $\mathbb E(T\wedge T_R)\leq(\tau R-1)/\sigma^2$. For $\tau\geq1$ this implies the weaker bound $\tau^2R/\sigma^2$. The threshold restriction prevents an impossible overshoot requirement below the starting value.
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