Sharp adaptive and pathwise stable similarity testing for scalar ergodic diffusions

Abstract: Within the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift

b

to some reference drift

b_{0}

: At prescribed significance, we simultaneously identify those regions where violation from similiarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise process. A detailed investigation for fractional driving noise, which is neither a semimartingale nor a Markov process, is provided for Hurst indices close to the Brownian motion case.

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