In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$. My question is how to find estimators for the process itself? So I want to estimate the process $\left(\sigma_{t}\right)_{t\geq 0}$.
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Based on high frequency data, it is not possible to estimate the process $\left(\sigma_{t}\right)_{t\geq 0}$. More precisely, there is no way to estimate the sign of it. Note that, if $W_{t}$ is a standard BM, then $-W_{t}$ is also a standard BM.
