= Fractional-moment control of Brownian increment ratios
{title2=$\mathbb E|R_\varepsilon/B_\varepsilon|^{1/4}\leq C(\mathbb E|R_\varepsilon|^2/\varepsilon)^{1/8}$}
The <Cauchy-Schwarz inequality> and <Jensen inequality> give the displayed estimate, with $C=(\mathbb E|N|^{-1/2})^{1/2}<\infty$. No independence between $R_\varepsilon$ and $B_\varepsilon$ is required. If $R_\varepsilon=\int_0^\varepsilon(H_s-H_0)dB_s$ for a bounded continuous adapted process, the <Itô isometry> and the <dominated convergence theorem> make the right side tend to zero. Time shifting proves recovery of $H_t$ in probability from the ratio of a short stochastic integral to its Brownian increment.
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