Sharp adaptive nonparametric testing for constant volatility

TL;DR

This paper introduces a minimax-optimal, adaptive nonparametric test for determining if the volatility function in a Gaussian white noise model is constant, based on discrete data.

Contribution

It develops a new testing procedure that is both minimax-optimal and adaptive for infill asymptotics, with a novel approach to measuring deviations from constancy.

Findings

Test is minimax-optimal and adaptive.

Deviations are measured via the ratio of $\sigma(t)$ to its $L^2$-average.

Constructs hypotheses with height $h(b)$ using solutions to $F_n(b)=0$.

Abstract

Based on discrete observations, we develop a test to infer if the volatility function $\sigma(\cdot)$ within the nonparametric Gaussian white noise model $dY_t = \sigma(t)dW_t$ 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.