Sharp adaptive nonparametric testing for constant volatility

Based on discrete observations, we develop a test to infer if the volatility function $\sigma (\cdot )$ within the nonparametric Gaussian white noise model $d{Y}_t=\sigma (t)d{W}_t$ is constant. The testing procedure is shown to be minimax-optimal and adaptive for infill asymptotics and these results entail that a deviation from the null hypothesis of constancy is best measured in terms of the ratio of $\sigma (t)$ and its $L^2$-average. The derivation of optimal constants requires the construction of hypotheses with height $h(b)$, where the parameter $b$ solves $F_n(b)=0$ for given functions $F_n$. Proving this equation to be solvable for each $n\in \mathbb{N}$ and establishing quantitative bounds of the solutions is built upon the implicit function theorem.