Fractional-moment control of Brownian increment ratios ($\mathbb{E}|R_{\epsilon }/B_{\epsilon }{|}^{1/4}\le C(\mathbb{E}|R_{\epsilon }{|}^2/\epsilon {)}^{1/8}$)

The Cauchy-Schwarz inequality and Jensen inequality give the displayed estimate, with $C=(\mathbb{E}|N{|}^{-1/2}{)}^{1/2}<\infty$. No independence between $R_{\epsilon }$ and $B_{\epsilon }$ is required. If $R_{\epsilon }={\int }_0^{\epsilon }(H_s-H_0)d{B}_s$ 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.